TPTP Problem File: ITP204^2.p
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%------------------------------------------------------------------------------
% File : ITP204^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer USubst problem prob_793__6341386_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 : USubst/prob_793__6341386_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 338 ( 111 unt; 51 typ; 0 def)
% Number of atoms : 739 ( 338 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 5346 ( 110 ~; 25 |; 53 &;4816 @)
% ( 0 <=>; 342 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 955 ( 955 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 46 usr; 7 con; 0-5 aty)
% Number of variables : 1181 ( 268 ^; 867 !; 15 ?;1181 :)
% ( 31 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:20:54.337
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Syntax_Ovariable,type,
variable: $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Syntax_Ogame,type,
game: $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_Syntax_Otrm,type,
trm: $tType ).
thf(ty_t_Syntax_Ofml,type,
fml: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
% Explicit typings (43)
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_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[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_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > 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_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Othe,type,
the:
!>[A: $tType] : ( ( option @ A ) > 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_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Static__Semantics_OFVF,type,
static_FVF: fml > ( set @ variable ) ).
thf(sy_c_Static__Semantics_OFVT,type,
static_FVT: trm > ( set @ variable ) ).
thf(sy_c_Syntax_Ofml_OGeq,type,
geq: trm > trm > fml ).
thf(sy_c_Syntax_Ofml_OPred,type,
pred: char > trm > fml ).
thf(sy_c_Syntax_Otrm_OConst,type,
const: char > trm ).
thf(sy_c_Syntax_Otrm_OFunc,type,
func: char > trm > trm ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OGeqo,type,
uSubst1556497037e_Geqo: ( option @ trm ) > ( option @ trm ) > ( option @ fml ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_Odotsubstt,type,
uSubst969145931substt: trm > ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_Ousappconst,type,
uSubst1138577137pconst: ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) > ( set @ variable ) > char > ( option @ trm ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_Ousubstappf,type,
uSubst95898978stappf: ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) > ( set @ variable ) > fml > ( option @ fml ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_Ousubstappt,type,
uSubst95898992stappt: ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) > ( set @ variable ) > trm > ( option @ trm ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_Ua____,type,
ua: set @ variable ).
thf(sy_v_Va____,type,
va: set @ variable ).
thf(sy_v__092_060sigma_062,type,
sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ).
thf(sy_v__092_060theta_062____,type,
theta: trm ).
thf(sy_v_p____,type,
p: char ).
thf(sy_v_vv____,type,
vv: ( set @ variable ) > ( set @ variable ) > variable ).
thf(sy_v_vva____,type,
vva: ( set @ variable ) > ( set @ variable ) > variable ).
% Relevant facts (255)
thf(fact_0__092_060open_062_092_060forall_062x0_Ax1_O_A_I_092_060exists_062v2_O_Av2_A_092_060in_062_Ax1_A_092_060and_062_A_I_092_060exists_062v3_O_Av3_A_092_060in_062_Ax0_A_092_060and_062_Av2_A_061_Av3_J_J_A_061_A_Ivv_Ax0_Ax1_A_092_060in_062_Ax1_A_092_060and_062_A_I_092_060exists_062v3_O_Av3_A_092_060in_062_Ax0_A_092_060and_062_Avv_Ax0_Ax1_A_061_Av3_J_J_092_060close_062,axiom,
! [X0: set @ variable,X1: set @ variable] :
( ( ? [V2: variable] :
( ( member @ variable @ V2 @ X1 )
& ? [V3: variable] :
( ( member @ variable @ V3 @ X0 )
& ( V2 = V3 ) ) ) )
= ( ( member @ variable @ ( vv @ X0 @ X1 ) @ X1 )
& ? [V3: variable] :
( ( member @ variable @ V3 @ X0 )
& ( ( vv @ X0 @ X1 )
= V3 ) ) ) ) ).
% \<open>\<forall>x0 x1. (\<exists>v2. v2 \<in> x1 \<and> (\<exists>v3. v3 \<in> x0 \<and> v2 = v3)) = (vv x0 x1 \<in> x1 \<and> (\<exists>v3. v3 \<in> x0 \<and> vv x0 x1 = v3))\<close>
thf(fact_1_Pred_Oprems_I2_J,axiom,
( ( uSubst95898978stappf @ sigma @ ua @ ( pred @ p @ theta ) )
!= ( none @ fml ) ) ).
% Pred.prems(2)
thf(fact_2__092_060open_062_Iif_AFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_A_092_060inter_062_AV_A_061_A_123_125_Athen_Ausubstappf_A_Idotsubstt_A_Ithe_A_Iusubstappt_A_092_060sigma_062_AV_A_092_060theta_062_J_J_J_A_123_125_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_Aelse_Aundeff_J_A_092_060noteq_062_Aundeff_092_060close_062,axiom,
~ ( ( ( inf_inf @ ( set @ variable )
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) )
@ va )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst95898978stappf @ ( uSubst969145931substt @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) @ ( bot_bot @ ( set @ variable ) )
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) )
= ( none @ fml ) ) ) ).
% \<open>(if FVF (the (SPreds \<sigma> p)) \<inter> V = {} then usubstappf (dotsubstt (the (usubstappt \<sigma> V \<theta>))) {} (the (SPreds \<sigma> p)) else undeff) \<noteq> undeff\<close>
thf(fact_3__092_060open_062_Iif_AFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_A_092_060inter_062_AV_A_061_A_123_125_Athen_Ausubstappf_A_Idotsubstt_A_Ithe_A_Iusubstappt_A_092_060sigma_062_AV_A_092_060theta_062_J_J_J_A_123_125_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_Aelse_Aundeff_J_A_092_060noteq_062_Ausubstappf_A_092_060sigma_062_AV_A_IPred_Ap_A_092_060theta_062_J_A_092_060Longrightarrow_062_ASPreds_A_092_060sigma_062_Ap_A_061_Aundeff_092_060close_062,axiom,
( ~ ( ( ( ( inf_inf @ ( set @ variable )
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) )
@ va )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst95898978stappf @ ( uSubst969145931substt @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) @ ( bot_bot @ ( set @ variable ) )
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) )
= ( uSubst95898978stappf @ sigma @ va @ ( pred @ p @ theta ) ) ) )
& ( ( ( inf_inf @ ( set @ variable )
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) )
@ va )
!= ( bot_bot @ ( set @ variable ) ) )
=> ( ( none @ fml )
= ( uSubst95898978stappf @ sigma @ va @ ( pred @ p @ theta ) ) ) ) )
=> ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p )
= ( none @ fml ) ) ) ).
% \<open>(if FVF (the (SPreds \<sigma> p)) \<inter> V = {} then usubstappf (dotsubstt (the (usubstappt \<sigma> V \<theta>))) {} (the (SPreds \<sigma> p)) else undeff) \<noteq> usubstappf \<sigma> V (Pred p \<theta>) \<Longrightarrow> SPreds \<sigma> p = undeff\<close>
thf(fact_4_usubstappf__det,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Phi: fml,V: set @ variable] :
( ( ( uSubst95898978stappf @ Sigma @ U @ Phi )
!= ( none @ fml ) )
=> ( ( ( uSubst95898978stappf @ Sigma @ V @ Phi )
!= ( none @ fml ) )
=> ( ( uSubst95898978stappf @ Sigma @ U @ Phi )
= ( uSubst95898978stappf @ Sigma @ V @ Phi ) ) ) ) ).
% usubstappf_det
thf(fact_5_f8,axiom,
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p )
= ( none @ fml ) )
=> ( ( uSubst95898978stappf @ sigma @ ua @ ( pred @ p @ theta ) )
= ( uSubst95898978stappf @ sigma @ va @ ( pred @ p @ theta ) ) ) ) ).
% f8
thf(fact_6__092_060open_062vv_AV_A_IFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_J_A_092_060notin_062_AFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_A_092_060or_062_Avva_AV_A_IFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_J_A_092_060notin_062_AV_A_092_060or_062_Avv_AV_A_IFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_J_A_092_060noteq_062_Avva_AV_A_IFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_J_092_060close_062,axiom,
( ~ ( member @ variable
@ ( vv @ va
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) )
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) )
| ~ ( member @ variable
@ ( vva @ va
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) )
@ va )
| ( ( vv @ va
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) )
!= ( vva @ va
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) ) ) ) ).
% \<open>vv V (FVF (the (SPreds \<sigma> p))) \<notin> FVF (the (SPreds \<sigma> p)) \<or> vva V (FVF (the (SPreds \<sigma> p))) \<notin> V \<or> vv V (FVF (the (SPreds \<sigma> p))) \<noteq> vva V (FVF (the (SPreds \<sigma> p)))\<close>
thf(fact_7_fml_Oinject_I1_J,axiom,
! [X11: char,X12: trm,Y11: char,Y12: trm] :
( ( ( pred @ X11 @ X12 )
= ( pred @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% fml.inject(1)
thf(fact_8_f7,axiom,
( ( uSubst95898992stappt @ sigma @ ua @ theta )
= ( uSubst95898992stappt @ sigma @ va @ theta ) ) ).
% f7
thf(fact_9_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F: B > C > D > A,X: product_prod @ B @ C,Y: D] :
( product_case_prod @ B @ C @ A
@ ^ [L: B,R: C] : ( F @ L @ R @ Y )
@ X ) ) ) ).
% case_prod_app
thf(fact_10_f1,axiom,
! [V4: variable] :
( ~ ( member @ variable @ V4 @ va )
| ( member @ variable @ V4 @ ua ) ) ).
% f1
thf(fact_11_f10,axiom,
( ( ( ( inf_inf @ ( set @ variable )
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) )
@ va )
!= ( bot_bot @ ( set @ variable ) ) )
| ! [V4: variable] :
( ~ ( member @ variable @ V4
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) )
| ! [Va: variable] :
( ~ ( member @ variable @ Va @ va )
| ( V4 != Va ) ) ) )
& ( ( ( inf_inf @ ( set @ variable )
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) )
@ va )
= ( bot_bot @ ( set @ variable ) ) )
| ( ( member @ variable
@ ( vv @ va
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) )
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) )
& ( member @ variable
@ ( vva @ va
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) )
@ va )
& ( ( vv @ va
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) )
= ( vva @ va
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) ) ) ) ) ) ).
% f10
thf(fact_12_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F2: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F2 @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X13: A,X2: B] : ( H @ ( F2 @ X13 @ X2 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_13_usubstappf__pred2,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),P2: char,R2: fml,U: set @ variable,Theta: trm] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ Sigma
@ P2 )
= ( some @ fml @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ R2 ) @ U )
!= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst95898978stappf @ Sigma @ U @ ( pred @ P2 @ Theta ) )
= ( none @ fml ) ) ) ) ).
% usubstappf_pred2
thf(fact_14_f2,axiom,
! [Z: option @ fml,F3: fml > ( option @ fml ),Za: option @ fml] :
( ( ( Za
= ( none @ fml ) )
=> ( ( case_option @ ( option @ fml ) @ fml @ Z @ F3 @ Za )
= Z ) )
& ( ( Za
!= ( none @ fml ) )
=> ( ( case_option @ ( option @ fml ) @ fml @ Z @ F3 @ Za )
= ( F3 @ ( the @ fml @ Za ) ) ) ) ) ).
% f2
thf(fact_15_Pred_Oprems_I1_J,axiom,
ord_less_eq @ ( set @ variable ) @ va @ ua ).
% Pred.prems(1)
thf(fact_16_f9,axiom,
! [V5: set @ variable,Va2: set @ variable] :
( ( ( ( inf_inf @ ( set @ variable ) @ V5 @ Va2 )
!= ( bot_bot @ ( set @ variable ) ) )
| ! [V4: variable] :
( ~ ( member @ variable @ V4 @ V5 )
| ! [Va: variable] :
( ~ ( member @ variable @ Va @ Va2 )
| ( V4 != Va ) ) ) )
& ( ( ( inf_inf @ ( set @ variable ) @ V5 @ Va2 )
= ( bot_bot @ ( set @ variable ) ) )
| ( ( member @ variable @ ( vv @ Va2 @ V5 ) @ V5 )
& ( member @ variable @ ( vva @ Va2 @ V5 ) @ Va2 )
& ( ( vv @ Va2 @ V5 )
= ( vva @ Va2 @ V5 ) ) ) ) ) ).
% f9
thf(fact_17__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062vva_O_A_092_060forall_062V_AVa_O_A_IV_A_092_060inter_062_AVa_A_092_060noteq_062_A_123_125_A_092_060or_062_A_I_092_060forall_062v_O_Av_A_092_060notin_062_AV_A_092_060or_062_A_I_092_060forall_062va_O_Ava_A_092_060notin_062_AVa_A_092_060or_062_Av_A_092_060noteq_062_Ava_J_J_J_A_092_060and_062_A_IV_A_092_060inter_062_AVa_A_061_A_123_125_A_092_060or_062_Avv_AVa_AV_A_092_060in_062_AV_A_092_060and_062_Avva_AVa_AV_A_092_060in_062_AVa_A_092_060and_062_Avv_AVa_AV_A_061_Avva_AVa_AV_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Vva: ( set @ variable ) > ( set @ variable ) > variable] :
~ ! [V5: set @ variable,Va2: set @ variable] :
( ( ( ( inf_inf @ ( set @ variable ) @ V5 @ Va2 )
!= ( bot_bot @ ( set @ variable ) ) )
| ! [V4: variable] :
( ~ ( member @ variable @ V4 @ V5 )
| ! [Va: variable] :
( ~ ( member @ variable @ Va @ Va2 )
| ( V4 != Va ) ) ) )
& ( ( ( inf_inf @ ( set @ variable ) @ V5 @ Va2 )
= ( bot_bot @ ( set @ variable ) ) )
| ( ( member @ variable @ ( vv @ Va2 @ V5 ) @ V5 )
& ( member @ variable @ ( Vva @ Va2 @ V5 ) @ Va2 )
& ( ( vv @ Va2 @ V5 )
= ( Vva @ Va2 @ V5 ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>vva. \<forall>V Va. (V \<inter> Va \<noteq> {} \<or> (\<forall>v. v \<notin> V \<or> (\<forall>va. va \<notin> Va \<or> v \<noteq> va))) \<and> (V \<inter> Va = {} \<or> vv Va V \<in> V \<and> vva Va V \<in> Va \<and> vv Va V = vva Va V) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_18_option_Oinject,axiom,
! [A: $tType,X22: A,Y2: A] :
( ( ( some @ A @ X22 )
= ( some @ A @ Y2 ) )
= ( X22 = Y2 ) ) ).
% option.inject
thf(fact_19_f5,axiom,
( ( uSubst95898992stappt @ sigma @ ua @ theta )
!= ( none @ trm ) ) ).
% f5
thf(fact_20_not__Some__eq,axiom,
! [A: $tType,X3: option @ A] :
( ( ! [Y: A] :
( X3
!= ( some @ A @ Y ) ) )
= ( X3
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_21_not__None__eq,axiom,
! [A: $tType,X3: option @ A] :
( ( X3
!= ( none @ A ) )
= ( ? [Y: A] :
( X3
= ( some @ A @ Y ) ) ) ) ).
% not_None_eq
thf(fact_22_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_23_option_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > B,X22: A] :
( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X22 ) )
= ( F22 @ X22 ) ) ).
% option.simps(5)
thf(fact_24_option_Osel,axiom,
! [A: $tType,X22: A] :
( ( the @ A @ ( some @ A @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_25_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P3: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P3 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P3 @ F1 ) )
& ( ( Option
= ( some @ A @ ( the @ A @ Option ) ) )
=> ( P3 @ ( F22 @ ( the @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_26_option_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H: B > C,F1: B,F22: A > B,Option: option @ A] :
( ( H @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( case_option @ C @ A @ ( H @ F1 )
@ ^ [X: A] : ( H @ ( F22 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_27_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P3: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P3 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P3 @ F1 ) )
| ( ( Option
= ( some @ A @ ( the @ A @ Option ) ) )
& ~ ( P3 @ ( F22 @ ( the @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_28_usubstappt__det,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,V: set @ variable] :
( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ V @ Theta )
!= ( none @ trm ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
= ( uSubst95898992stappt @ Sigma @ V @ Theta ) ) ) ) ).
% usubstappt_det
thf(fact_29_usubstappt__antimon,axiom,
! [V: set @ variable,U: set @ variable,Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Theta: trm] :
( ( ord_less_eq @ ( set @ variable ) @ V @ U )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
= ( uSubst95898992stappt @ Sigma @ V @ Theta ) ) ) ) ).
% usubstappt_antimon
thf(fact_30_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_31_combine__options__cases,axiom,
! [A: $tType,B: $tType,X3: option @ A,P3: ( option @ A ) > ( option @ B ) > $o,Y3: option @ B] :
( ( ( X3
= ( none @ A ) )
=> ( P3 @ X3 @ Y3 ) )
=> ( ( ( Y3
= ( none @ B ) )
=> ( P3 @ X3 @ Y3 ) )
=> ( ! [A2: A,B2: B] :
( ( X3
= ( some @ A @ A2 ) )
=> ( ( Y3
= ( some @ B @ B2 ) )
=> ( P3 @ X3 @ Y3 ) ) )
=> ( P3 @ X3 @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_32_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P4: ( option @ A ) > $o] :
! [X4: option @ A] : ( P4 @ X4 ) )
= ( ^ [P: ( option @ A ) > $o] :
( ( P @ ( none @ A ) )
& ! [X: A] : ( P @ ( some @ A @ X ) ) ) ) ) ).
% split_option_all
thf(fact_33_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P4: ( option @ A ) > $o] :
? [X4: option @ A] : ( P4 @ X4 ) )
= ( ^ [P: ( option @ A ) > $o] :
( ( P @ ( none @ A ) )
| ? [X: A] : ( P @ ( some @ A @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_34_option_Oinducts,axiom,
! [A: $tType,P3: ( option @ A ) > $o,Option: option @ A] :
( ( P3 @ ( none @ A ) )
=> ( ! [X5: A] : ( P3 @ ( some @ A @ X5 ) )
=> ( P3 @ Option ) ) ) ).
% option.inducts
thf(fact_35_option_Oexhaust,axiom,
! [A: $tType,Y3: option @ A] :
( ( Y3
!= ( none @ A ) )
=> ~ ! [X23: A] :
( Y3
!= ( some @ A @ X23 ) ) ) ).
% option.exhaust
thf(fact_36_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X22: A] :
( ( Option
= ( some @ A @ X22 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_37_option_Odistinct_I1_J,axiom,
! [A: $tType,X22: A] :
( ( none @ A )
!= ( some @ A @ X22 ) ) ).
% option.distinct(1)
thf(fact_38_option_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_option @ B @ A )
= ( ^ [F12: B,F23: A > B,Option2: option @ A] :
( if @ B
@ ( Option2
= ( none @ A ) )
@ F12
@ ( F23 @ ( the @ A @ Option2 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_39_option_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > B] :
( ( case_option @ B @ A @ F1 @ F22 @ ( none @ A ) )
= F1 ) ).
% option.simps(4)
thf(fact_40_Existso_Oinduct,axiom,
! [P3: variable > ( option @ fml ) > $o,A0: variable,A1: option @ fml] :
( ! [X5: variable,Phi2: fml] : ( P3 @ X5 @ ( some @ fml @ Phi2 ) )
=> ( ! [X5: variable] : ( P3 @ X5 @ ( none @ fml ) )
=> ( P3 @ A0 @ A1 ) ) ) ).
% Existso.induct
thf(fact_41_undeff__equiv,axiom,
! [Phi: option @ fml] :
( ( Phi
!= ( none @ fml ) )
= ( ? [F: fml] :
( Phi
= ( some @ fml @ F ) ) ) ) ).
% undeff_equiv
thf(fact_42_Testo_Oinduct,axiom,
! [P3: ( option @ fml ) > $o,A0: option @ fml] :
( ! [Phi2: fml] : ( P3 @ ( some @ fml @ Phi2 ) )
=> ( ( P3 @ ( none @ fml ) )
=> ( P3 @ A0 ) ) ) ).
% Testo.induct
thf(fact_43_Testo_Ocases,axiom,
! [X3: option @ fml] :
( ! [Phi2: fml] :
( X3
!= ( some @ fml @ Phi2 ) )
=> ( X3
= ( none @ fml ) ) ) ).
% Testo.cases
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P3: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P3 ) )
= ( P3 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P3 @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P3 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X5: A] :
( ( F2 @ X5 )
= ( G @ X5 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_48_Ando_Oinduct,axiom,
! [P3: ( option @ fml ) > ( option @ fml ) > $o,A0: option @ fml,A1: option @ fml] :
( ! [Phi2: fml,Psi: fml] : ( P3 @ ( some @ fml @ Phi2 ) @ ( some @ fml @ Psi ) )
=> ( ! [X_1: option @ fml] : ( P3 @ ( none @ fml ) @ X_1 )
=> ( ! [V6: fml] : ( P3 @ ( some @ fml @ V6 ) @ ( none @ fml ) )
=> ( P3 @ A0 @ A1 ) ) ) ) ).
% Ando.induct
thf(fact_49_option_Oexpand,axiom,
! [A: $tType,Option: option @ A,Option3: option @ A] :
( ( ( Option
= ( none @ A ) )
= ( Option3
= ( none @ A ) ) )
=> ( ( ( Option
!= ( none @ A ) )
=> ( ( Option3
!= ( none @ A ) )
=> ( ( the @ A @ Option )
= ( the @ A @ Option3 ) ) ) )
=> ( Option = Option3 ) ) ) ).
% option.expand
thf(fact_50_usubstappf__pred__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,P2: char,Theta: trm] :
( ( ( uSubst95898978stappf @ Sigma @ U @ ( pred @ P2 @ Theta ) )
!= ( none @ fml ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
& ( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ Sigma
@ P2 )
= ( none @ fml ) )
| ? [R3: fml] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ Sigma
@ P2 )
= ( some @ fml @ R3 ) )
& ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ R3 ) @ U )
= ( bot_bot @ ( set @ variable ) ) ) ) ) ) ) ).
% usubstappf_pred_conv
thf(fact_51_f6,axiom,
( ( case_option @ ( option @ fml ) @ trm @ ( none @ fml )
@ ^ [T: trm] :
( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ p @ T ) )
@ ^ [F: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ F ) @ ua )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ T ) @ ( bot_bot @ ( set @ variable ) ) @ F )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) )
@ ( uSubst95898992stappt @ sigma @ ua @ theta ) )
= ( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ p @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ ua @ theta ) ) ) )
@ ^ [F: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ F ) @ ua )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ ua @ theta ) ) ) @ ( bot_bot @ ( set @ variable ) ) @ F )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) ).
% f6
thf(fact_52_f3,axiom,
( ( case_option @ ( option @ fml ) @ trm @ ( none @ fml )
@ ^ [T: trm] :
( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ p @ T ) )
@ ^ [F: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ F ) @ ua )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ T ) @ ( bot_bot @ ( set @ variable ) ) @ F )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) )
@ ( uSubst95898992stappt @ sigma @ ua @ theta ) )
!= ( none @ fml ) ) ).
% f3
thf(fact_53_usubstappf_Osimps_I1_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,P2: char,Theta: trm] :
( ( uSubst95898978stappf @ Sigma @ U @ ( pred @ P2 @ Theta ) )
= ( case_option @ ( option @ fml ) @ trm @ ( none @ fml )
@ ^ [Sigma_theta: trm] :
( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ P2 @ Sigma_theta ) )
@ ^ [R: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ R ) @ U )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ Sigma_theta ) @ ( bot_bot @ ( set @ variable ) ) @ R )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ Sigma
@ P2 ) )
@ ( uSubst95898992stappt @ Sigma @ U @ Theta ) ) ) ).
% usubstappf.simps(1)
thf(fact_54_Int__subset__iff,axiom,
! [A: $tType,C2: set @ A,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C2 @ A4 )
& ( ord_less_eq @ ( set @ A ) @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_55_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X3 )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_56_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_57_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_58_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_59_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z2 ) )
= ( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% le_inf_iff
thf(fact_60_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B4: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B4 @ C3 ) )
= ( ( ord_less_eq @ A @ A3 @ B4 )
& ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% inf.bounded_iff
thf(fact_61_f4,axiom,
! [Z: option @ fml,F3: trm > ( option @ fml ),Za: option @ trm] :
( ( ( Za
= ( none @ trm ) )
=> ( ( case_option @ ( option @ fml ) @ trm @ Z @ F3 @ Za )
= Z ) )
& ( ( Za
!= ( none @ trm ) )
=> ( ( case_option @ ( option @ fml ) @ trm @ Z @ F3 @ Za )
= ( F3 @ ( the @ trm @ Za ) ) ) ) ) ).
% f4
thf(fact_62_empty__Collect__eq,axiom,
! [A: $tType,P3: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P3 ) )
= ( ! [X: A] :
~ ( P3 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_63_Collect__empty__eq,axiom,
! [A: $tType,P3: A > $o] :
( ( ( collect @ A @ P3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
~ ( P3 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_64_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X: A] :
~ ( member @ A @ X @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_65_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_66_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ).
% subset_antisym
thf(fact_67_subsetI,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( member @ A @ X5 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% subsetI
thf(fact_68_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ Y3 )
= ( inf_inf @ A @ X3 @ Y3 ) ) ) ).
% inf_right_idem
thf(fact_69_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B4: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B4 ) @ B4 )
= ( inf_inf @ A @ A3 @ B4 ) ) ) ).
% inf.right_idem
thf(fact_70_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ X3 @ Y3 ) )
= ( inf_inf @ A @ X3 @ Y3 ) ) ) ).
% inf_left_idem
thf(fact_71_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B4: A] :
( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B4 ) )
= ( inf_inf @ A @ A3 @ B4 ) ) ) ).
% inf.left_idem
thf(fact_72_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A] :
( ( inf_inf @ A @ X3 @ X3 )
= X3 ) ) ).
% inf_idem
thf(fact_73_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ A3 )
= A3 ) ) ).
% inf.idem
thf(fact_74_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F: A > B,G2: A > B,X: A] : ( inf_inf @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ) ).
% inf_apply
thf(fact_75_Int__iff,axiom,
! [A: $tType,C3: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( ( member @ A @ C3 @ A4 )
& ( member @ A @ C3 @ B3 ) ) ) ).
% Int_iff
thf(fact_76_IntI,axiom,
! [A: $tType,C3: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C3 @ A4 )
=> ( ( member @ A @ C3 @ B3 )
=> ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% IntI
thf(fact_77_usubstappf__pred,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),P2: char,R2: fml,U: set @ variable,Theta: trm,Sigma_theta2: trm] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ Sigma
@ P2 )
= ( some @ fml @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ R2 ) @ U )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
= ( some @ trm @ Sigma_theta2 ) )
=> ( ( uSubst95898978stappf @ Sigma @ U @ ( pred @ P2 @ Theta ) )
= ( uSubst95898978stappf @ ( uSubst969145931substt @ Sigma_theta2 ) @ ( bot_bot @ ( set @ variable ) ) @ R2 ) ) ) ) ) ).
% usubstappf_pred
thf(fact_78_ODEo_Oinduct,axiom,
! [P3: char > ( option @ trm ) > $o,A0: char,A1: option @ trm] :
( ! [X5: char,Theta2: trm] : ( P3 @ X5 @ ( some @ trm @ Theta2 ) )
=> ( ! [X5: char] : ( P3 @ X5 @ ( none @ trm ) )
=> ( P3 @ A0 @ A1 ) ) ) ).
% ODEo.induct
thf(fact_79_undeft__None,axiom,
( ( none @ trm )
= ( none @ trm ) ) ).
% undeft_None
thf(fact_80_undeft__equiv,axiom,
! [Theta: option @ trm] :
( ( Theta
!= ( none @ trm ) )
= ( ? [T: trm] :
( Theta
= ( some @ trm @ T ) ) ) ) ).
% undeft_equiv
thf(fact_81_Timeso_Oinduct,axiom,
! [P3: ( option @ trm ) > ( option @ trm ) > $o,A0: option @ trm,A1: option @ trm] :
( ! [Theta2: trm,Eta: trm] : ( P3 @ ( some @ trm @ Theta2 ) @ ( some @ trm @ Eta ) )
=> ( ! [X_1: option @ trm] : ( P3 @ ( none @ trm ) @ X_1 )
=> ( ! [V6: trm] : ( P3 @ ( some @ trm @ V6 ) @ ( none @ trm ) )
=> ( P3 @ A0 @ A1 ) ) ) ) ).
% Timeso.induct
thf(fact_82_Assigno_Oinduct,axiom,
! [P3: variable > ( option @ trm ) > $o,A0: variable,A1: option @ trm] :
( ! [X5: variable,Theta2: trm] : ( P3 @ X5 @ ( some @ trm @ Theta2 ) )
=> ( ! [X5: variable] : ( P3 @ X5 @ ( none @ trm ) )
=> ( P3 @ A0 @ A1 ) ) ) ).
% Assigno.induct
thf(fact_83_Differentialo_Ocases,axiom,
! [X3: option @ trm] :
( ! [Theta2: trm] :
( X3
!= ( some @ trm @ Theta2 ) )
=> ( X3
= ( none @ trm ) ) ) ).
% Differentialo.cases
thf(fact_84_Differentialo_Oinduct,axiom,
! [P3: ( option @ trm ) > $o,A0: option @ trm] :
( ! [Theta2: trm] : ( P3 @ ( some @ trm @ Theta2 ) )
=> ( ( P3 @ ( none @ trm ) )
=> ( P3 @ A0 ) ) ) ).
% Differentialo.induct
thf(fact_85_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_86_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A5 )
@ ^ [X: A] : ( member @ A @ X @ B5 ) ) ) ) ) ).
% inf_set_def
thf(fact_87_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A5 )
@ ^ [X: A] : ( member @ A @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_88_option_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
= ( case_option @ $o @ A @ $false
@ ^ [Uu: A] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_89_option_Odisc__eq__case_I1_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
= ( none @ A ) )
= ( case_option @ $o @ A @ $true
@ ^ [Uu: A] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_90_Diamondo_Oinduct,axiom,
! [P3: ( option @ game ) > ( option @ fml ) > $o,A0: option @ game,A1: option @ fml] :
( ! [Alpha: game,Phi2: fml] : ( P3 @ ( some @ game @ Alpha ) @ ( some @ fml @ Phi2 ) )
=> ( ! [X_1: option @ fml] : ( P3 @ ( none @ game ) @ X_1 )
=> ( ! [V6: game] : ( P3 @ ( some @ game @ V6 ) @ ( none @ fml ) )
=> ( P3 @ A0 @ A1 ) ) ) ) ).
% Diamondo.induct
thf(fact_91_case__optionE,axiom,
! [A: $tType,P3: $o,Q: A > $o,X3: option @ A] :
( ( case_option @ $o @ A @ P3 @ Q @ X3 )
=> ( ( ( X3
= ( none @ A ) )
=> ~ P3 )
=> ~ ! [Y4: A] :
( ( X3
= ( some @ A @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_92_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X: A] : ( member @ A @ X @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_93_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_94_equals0D,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A4 ) ) ).
% equals0D
thf(fact_95_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_96_Collect__mono__iff,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P3 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_97_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z3: set @ A] : Y5 = Z3 )
= ( ^ [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_98_subset__trans,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_99_Collect__mono,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P3 @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_100_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_101_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [T: A] :
( ( member @ A @ T @ A5 )
=> ( member @ A @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_102_equalityD2,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( A4 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A4 ) ) ).
% equalityD2
thf(fact_103_equalityD1,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( A4 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B3 ) ) ).
% equalityD1
thf(fact_104_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_105_equalityE,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( A4 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A4 ) ) ) ).
% equalityE
thf(fact_106_subsetD,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( member @ A @ C3 @ A4 )
=> ( member @ A @ C3 @ B3 ) ) ) ).
% subsetD
thf(fact_107_in__mono,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,X3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B3 ) ) ) ).
% in_mono
thf(fact_108_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z2 ) )
= ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X3 @ Z2 ) ) ) ) ).
% inf_left_commute
thf(fact_109_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B4: A,A3: A,C3: A] :
( ( inf_inf @ A @ B4 @ ( inf_inf @ A @ A3 @ C3 ) )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B4 @ C3 ) ) ) ) ).
% inf.left_commute
thf(fact_110_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X: A,Y: A] : ( inf_inf @ A @ Y @ X ) ) ) ) ).
% inf_commute
thf(fact_111_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A6: A,B6: A] : ( inf_inf @ A @ B6 @ A6 ) ) ) ) ).
% inf.commute
thf(fact_112_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ Z2 )
= ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z2 ) ) ) ) ).
% inf_assoc
thf(fact_113_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B4: A,C3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B4 ) @ C3 )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B4 @ C3 ) ) ) ) ).
% inf.assoc
thf(fact_114_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B3: A,K: A,B4: A,A3: A] :
( ( B3
= ( inf_inf @ A @ K @ B4 ) )
=> ( ( inf_inf @ A @ A3 @ B3 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B4 ) ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_115_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: A,K: A,A3: A,B4: A] :
( ( A4
= ( inf_inf @ A @ K @ A3 ) )
=> ( ( inf_inf @ A @ A4 @ B4 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B4 ) ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_116_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F: A > B,G2: A > B,X: A] : ( inf_inf @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ) ).
% inf_fun_def
thf(fact_117_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( inf_inf @ A )
= ( ^ [X: A,Y: A] : ( inf_inf @ A @ Y @ X ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_118_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ Z2 )
= ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z2 ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_119_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z2 ) )
= ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X3 @ Z2 ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_120_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y3: A] :
( ( inf_inf @ A @ X3 @ ( inf_inf @ A @ X3 @ Y3 ) )
= ( inf_inf @ A @ X3 @ Y3 ) ) ) ).
% inf_sup_aci(4)
thf(fact_121_Int__left__commute,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) )
= ( inf_inf @ ( set @ A ) @ B3 @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_122_Int__left__absorb,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ).
% Int_left_absorb
thf(fact_123_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_124_Int__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Int_absorb
thf(fact_125_Int__assoc,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_126_IntD2,axiom,
! [A: $tType,C3: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
=> ( member @ A @ C3 @ B3 ) ) ).
% IntD2
thf(fact_127_IntD1,axiom,
! [A: $tType,C3: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
=> ( member @ A @ C3 @ A4 ) ) ).
% IntD1
thf(fact_128_IntE,axiom,
! [A: $tType,C3: A,A4: set @ A,B3: set @ A] :
( ( member @ A @ C3 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) )
=> ~ ( ( member @ A @ C3 @ A4 )
=> ~ ( member @ A @ C3 @ B3 ) ) ) ).
% IntE
thf(fact_129_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X: A] : $false ) ) ).
% empty_def
thf(fact_130_Collect__subset,axiom,
! [A: $tType,A4: set @ A,P3: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A4 )
& ( P3 @ X ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_131_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A5 )
& ( member @ A @ X @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_132_Int__Collect,axiom,
! [A: $tType,X3: A,A4: set @ A,P3: A > $o] :
( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P3 ) ) )
= ( ( member @ A @ X3 @ A4 )
& ( P3 @ X3 ) ) ) ).
% Int_Collect
thf(fact_133_Collect__conj__eq,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( P3 @ X )
& ( Q @ X ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_134_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B4: A,C3: A,A3: A] :
( ( ord_less_eq @ A @ B4 @ C3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B4 ) @ C3 ) ) ) ).
% inf.coboundedI2
thf(fact_135_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C3: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B4 ) @ C3 ) ) ) ).
% inf.coboundedI1
thf(fact_136_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A6: A] :
( ( inf_inf @ A @ A6 @ B6 )
= B6 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_137_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B6: A] :
( ( inf_inf @ A @ A6 @ B6 )
= A6 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_138_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B4: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B4 ) @ B4 ) ) ).
% inf.cobounded2
thf(fact_139_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B4: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B4 ) @ A3 ) ) ).
% inf.cobounded1
thf(fact_140_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B6: A] :
( A6
= ( inf_inf @ A @ A6 @ B6 ) ) ) ) ) ).
% inf.order_iff
thf(fact_141_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z2 ) ) ) ) ) ).
% inf_greatest
thf(fact_142_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B4: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B4 @ C3 ) ) ) ) ) ).
% inf.boundedI
thf(fact_143_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B4: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B4 @ C3 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B4 )
=> ~ ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% inf.boundedE
thf(fact_144_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( inf_inf @ A @ X3 @ Y3 )
= Y3 ) ) ) ).
% inf_absorb2
thf(fact_145_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( inf_inf @ A @ X3 @ Y3 )
= X3 ) ) ) ).
% inf_absorb1
thf(fact_146_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B4: A,A3: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B4 )
= B4 ) ) ) ).
% inf.absorb2
thf(fact_147_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( inf_inf @ A @ A3 @ B4 )
= A3 ) ) ) ).
% inf.absorb1
thf(fact_148_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y: A] :
( ( inf_inf @ A @ X @ Y )
= X ) ) ) ) ).
% le_iff_inf
thf(fact_149_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F2: A > A > A,X3: A,Y3: A] :
( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ ( F2 @ X5 @ Y4 ) @ X5 )
=> ( ! [X5: A,Y4: A] : ( ord_less_eq @ A @ ( F2 @ X5 @ Y4 ) @ Y4 )
=> ( ! [X5: A,Y4: A,Z4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ( ord_less_eq @ A @ X5 @ Z4 )
=> ( ord_less_eq @ A @ X5 @ ( F2 @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf @ A @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ) ).
% inf_unique
thf(fact_150_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B4: A] :
( ( A3
= ( inf_inf @ A @ A3 @ B4 ) )
=> ( ord_less_eq @ A @ A3 @ B4 ) ) ) ).
% inf.orderI
thf(fact_151_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( A3
= ( inf_inf @ A @ A3 @ B4 ) ) ) ) ).
% inf.orderE
thf(fact_152_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B4: A,X3: A,A3: A] :
( ( ord_less_eq @ A @ B4 @ X3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B4 ) @ X3 ) ) ) ).
% le_infI2
thf(fact_153_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X3: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B4 ) @ X3 ) ) ) ).
% le_infI1
thf(fact_154_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C3: A,B4: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
=> ( ( ord_less_eq @ A @ B4 @ D2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B4 ) @ ( inf_inf @ A @ C3 @ D2 ) ) ) ) ) ).
% inf_mono
thf(fact_155_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,A3: A,B4: A] :
( ( ord_less_eq @ A @ X3 @ A3 )
=> ( ( ord_less_eq @ A @ X3 @ B4 )
=> ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A3 @ B4 ) ) ) ) ) ).
% le_infI
thf(fact_156_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,A3: A,B4: A] :
( ( ord_less_eq @ A @ X3 @ ( inf_inf @ A @ A3 @ B4 ) )
=> ~ ( ( ord_less_eq @ A @ X3 @ A3 )
=> ~ ( ord_less_eq @ A @ X3 @ B4 ) ) ) ) ).
% le_infE
thf(fact_157_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ Y3 ) ) ).
% inf_le2
thf(fact_158_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ X3 ) ) ).
% inf_le1
thf(fact_159_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ X3 ) ) ).
% inf_sup_ord(1)
thf(fact_160_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ Y3 ) ) ).
% inf_sup_ord(2)
thf(fact_161_disjoint__iff__not__equal,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ! [Y: A] :
( ( member @ A @ Y @ B3 )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_162_Int__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_163_Int__empty__left,axiom,
! [A: $tType,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_164_disjoint__iff,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A4 )
=> ~ ( member @ A @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_165_Int__emptyI,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ~ ( member @ A @ X5 @ B3 ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_166_Int__Collect__mono,axiom,
! [A: $tType,A4: set @ A,B3: set @ A,P3: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( ( P3 @ X5 )
=> ( Q @ X5 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P3 ) ) @ ( inf_inf @ ( set @ A ) @ B3 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_167_Int__greatest,axiom,
! [A: $tType,C2: set @ A,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ C2 @ B3 )
=> ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_168_Int__absorb2,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= A4 ) ) ).
% Int_absorb2
thf(fact_169_Int__absorb1,axiom,
! [A: $tType,B3: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_170_Int__lower2,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_171_Int__lower1,axiom,
! [A: $tType,A4: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ A4 ) ).
% Int_lower1
thf(fact_172_Int__mono,axiom,
! [A: $tType,A4: set @ A,C2: set @ A,B3: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B3 ) @ ( inf_inf @ ( set @ A ) @ C2 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_173_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_174_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_175_subset__emptyI,axiom,
! [A: $tType,A4: set @ A] :
( ! [X5: A] :
~ ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_176_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_177_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_178_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_179_usubstappf__geqr,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta2: trm] :
( ( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
!= ( none @ fml ) )
=> ( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
= ( some @ fml @ ( geq @ ( the @ trm @ ( uSubst95898992stappt @ Sigma @ U @ Theta ) ) @ ( the @ trm @ ( uSubst95898992stappt @ Sigma @ U @ Eta2 ) ) ) ) ) ) ).
% usubstappf_geqr
thf(fact_180_fml_Oinject_I2_J,axiom,
! [X21: trm,X222: trm,Y21: trm,Y22: trm] :
( ( ( geq @ X21 @ X222 )
= ( geq @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% fml.inject(2)
thf(fact_181_predicate1I,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P3 @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq @ ( A > $o ) @ P3 @ Q ) ) ).
% predicate1I
thf(fact_182_inf1I,axiom,
! [A: $tType,A4: A > $o,X3: A,B3: A > $o] :
( ( A4 @ X3 )
=> ( ( B3 @ X3 )
=> ( inf_inf @ ( A > $o ) @ A4 @ B3 @ X3 ) ) ) ).
% inf1I
thf(fact_183_inf1E,axiom,
! [A: $tType,A4: A > $o,B3: A > $o,X3: A] :
( ( inf_inf @ ( A > $o ) @ A4 @ B3 @ X3 )
=> ~ ( ( A4 @ X3 )
=> ~ ( B3 @ X3 ) ) ) ).
% inf1E
thf(fact_184_inf1D1,axiom,
! [A: $tType,A4: A > $o,B3: A > $o,X3: A] :
( ( inf_inf @ ( A > $o ) @ A4 @ B3 @ X3 )
=> ( A4 @ X3 ) ) ).
% inf1D1
thf(fact_185_inf1D2,axiom,
! [A: $tType,A4: A > $o,B3: A > $o,X3: A] :
( ( inf_inf @ ( A > $o ) @ A4 @ B3 @ X3 )
=> ( B3 @ X3 ) ) ).
% inf1D2
thf(fact_186_predicate1D,axiom,
! [A: $tType,P3: A > $o,Q: A > $o,X3: A] :
( ( ord_less_eq @ ( A > $o ) @ P3 @ Q )
=> ( ( P3 @ X3 )
=> ( Q @ X3 ) ) ) ).
% predicate1D
thf(fact_187_rev__predicate1D,axiom,
! [A: $tType,P3: A > $o,X3: A,Q: A > $o] :
( ( P3 @ X3 )
=> ( ( ord_less_eq @ ( A > $o ) @ P3 @ Q )
=> ( Q @ X3 ) ) ) ).
% rev_predicate1D
thf(fact_188_Aterm__Some,axiom,
( ( some @ trm )
= ( some @ trm ) ) ).
% Aterm_Some
thf(fact_189_Loopo_Ocases,axiom,
! [X3: option @ game] :
( ! [Alpha: game] :
( X3
!= ( some @ game @ Alpha ) )
=> ( X3
= ( none @ game ) ) ) ).
% Loopo.cases
thf(fact_190_Loopo_Oinduct,axiom,
! [P3: ( option @ game ) > $o,A0: option @ game] :
( ! [Alpha: game] : ( P3 @ ( some @ game @ Alpha ) )
=> ( ( P3 @ ( none @ game ) )
=> ( P3 @ A0 ) ) ) ).
% Loopo.induct
thf(fact_191_undefg__equiv,axiom,
! [Alpha2: option @ game] :
( ( Alpha2
!= ( none @ game ) )
= ( ? [G2: game] :
( Alpha2
= ( some @ game @ G2 ) ) ) ) ).
% undefg_equiv
thf(fact_192_Composeo_Oinduct,axiom,
! [P3: ( option @ game ) > ( option @ game ) > $o,A0: option @ game,A1: option @ game] :
( ! [Alpha: game,Beta: game] : ( P3 @ ( some @ game @ Alpha ) @ ( some @ game @ Beta ) )
=> ( ! [Alpha: option @ game] : ( P3 @ Alpha @ ( none @ game ) )
=> ( ! [V6: game] : ( P3 @ ( none @ game ) @ ( some @ game @ V6 ) )
=> ( P3 @ A0 @ A1 ) ) ) ) ).
% Composeo.induct
thf(fact_193_fml_Odistinct_I1_J,axiom,
! [X11: char,X12: trm,X21: trm,X222: trm] :
( ( pred @ X11 @ X12 )
!= ( geq @ X21 @ X222 ) ) ).
% fml.distinct(1)
thf(fact_194_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funD
thf(fact_195_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funE
thf(fact_196_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G: A > B] :
( ! [X5: A] : ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% le_funI
thf(fact_197_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_198_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B4: B,C3: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C3 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_199_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B4: A,F2: A > C,C3: C] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ C @ ( F2 @ B4 ) @ C3 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ C @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A3 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_200_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F2: B > A,B4: B,C3: B] :
( ( A3
= ( F2 @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C3 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F2 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_201_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B4: A,F2: A > B,C3: B] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ( F2 @ B4 )
= C3 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A3 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_202_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ).
% eq_iff
thf(fact_203_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ) ).
% antisym
thf(fact_204_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% linear
thf(fact_205_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( X3 = Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% eq_refl
thf(fact_206_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% le_cases
thf(fact_207_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B4: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% order.trans
thf(fact_208_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_209_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_210_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
& ( ord_less_eq @ A @ B6 @ A6 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_211_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B4: A,C3: A] :
( ( A3 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_212_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B4: A,C3: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( B4 = C3 )
=> ( ord_less_eq @ A @ A3 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_213_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_214_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_215_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_216_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > A > $o,A3: A,B4: A] :
( ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( P3 @ A2 @ B2 ) )
=> ( ! [A2: A,B2: A] :
( ( P3 @ B2 @ A2 )
=> ( P3 @ A2 @ B2 ) )
=> ( P3 @ A3 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_217_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A3: A,C3: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ( ord_less_eq @ A @ C3 @ B4 )
=> ( ord_less_eq @ A @ C3 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_218_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [A6: A,B6: A] :
( ( ord_less_eq @ A @ B6 @ A6 )
& ( ord_less_eq @ A @ A6 @ B6 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_219_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B4: A,A3: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B4 )
=> ( A3 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_220_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_221_prop__restrict,axiom,
! [A: $tType,X3: A,Z5: set @ A,X6: set @ A,P3: A > $o] :
( ( member @ A @ X3 @ Z5 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z5
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ X6 )
& ( P3 @ X ) ) ) )
=> ( P3 @ X3 ) ) ) ).
% prop_restrict
thf(fact_222_Collect__restrict,axiom,
! [A: $tType,X6: set @ A,P3: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ X6 )
& ( P3 @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_223_usubstappf__geq__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta2: trm] :
( ( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
!= ( none @ fml ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
& ( ( uSubst95898992stappt @ Sigma @ U @ Eta2 )
!= ( none @ trm ) ) ) ) ).
% usubstappf_geq_conv
thf(fact_224_usubstappf__geq,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta2: trm] :
( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Eta2 )
!= ( none @ trm ) )
=> ( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
= ( some @ fml @ ( geq @ ( the @ trm @ ( uSubst95898992stappt @ Sigma @ U @ Theta ) ) @ ( the @ trm @ ( uSubst95898992stappt @ Sigma @ U @ Eta2 ) ) ) ) ) ) ) ).
% usubstappf_geq
thf(fact_225_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P3: $o,Q: A > $o,X3: option @ A,R4: B > $o,F2: B,G: A > B] :
( ( case_option @ $o @ A @ P3 @ Q @ X3 )
=> ( ( P3
=> ( R4 @ F2 ) )
=> ( ! [Q2: A] :
( ( Q @ Q2 )
=> ( R4 @ ( G @ Q2 ) ) )
=> ( R4 @ ( case_option @ B @ A @ F2 @ G @ X3 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_226_Geqo_Oelims,axiom,
! [X3: option @ trm,Xa: option @ trm,Y3: option @ fml] :
( ( ( uSubst1556497037e_Geqo @ X3 @ Xa )
= Y3 )
=> ( ! [Theta2: trm] :
( ( X3
= ( some @ trm @ Theta2 ) )
=> ! [Eta: trm] :
( ( Xa
= ( some @ trm @ Eta ) )
=> ( Y3
!= ( some @ fml @ ( geq @ Theta2 @ Eta ) ) ) ) )
=> ( ( ( X3
= ( none @ trm ) )
=> ( Y3
!= ( none @ fml ) ) )
=> ~ ( ? [V6: trm] :
( X3
= ( some @ trm @ V6 ) )
=> ( ( Xa
= ( none @ trm ) )
=> ( Y3
!= ( none @ fml ) ) ) ) ) ) ) ).
% Geqo.elims
thf(fact_227_inf__Int__eq,axiom,
! [A: $tType,R4: set @ A,S: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ R4 )
@ ^ [X: A] : ( member @ A @ X @ S ) )
= ( ^ [X: A] : ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ R4 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_228_Geqo_Osimps_I2_J,axiom,
! [Eta2: option @ trm] :
( ( uSubst1556497037e_Geqo @ ( none @ trm ) @ Eta2 )
= ( none @ fml ) ) ).
% Geqo.simps(2)
thf(fact_229_Geqo__undef,axiom,
! [Theta: option @ trm,Eta2: option @ trm] :
( ( ( uSubst1556497037e_Geqo @ Theta @ Eta2 )
= ( none @ fml ) )
= ( ( Theta
= ( none @ trm ) )
| ( Eta2
= ( none @ trm ) ) ) ) ).
% Geqo_undef
thf(fact_230_Geqo_Osimps_I3_J,axiom,
! [V7: trm] :
( ( uSubst1556497037e_Geqo @ ( some @ trm @ V7 ) @ ( none @ trm ) )
= ( none @ fml ) ) ).
% Geqo.simps(3)
thf(fact_231_Geqo_Osimps_I1_J,axiom,
! [Theta: trm,Eta2: trm] :
( ( uSubst1556497037e_Geqo @ ( some @ trm @ Theta ) @ ( some @ trm @ Eta2 ) )
= ( some @ fml @ ( geq @ Theta @ Eta2 ) ) ) ).
% Geqo.simps(1)
thf(fact_232_usubstappf_Osimps_I2_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta2: trm] :
( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
= ( uSubst1556497037e_Geqo @ ( uSubst95898992stappt @ Sigma @ U @ Theta ) @ ( uSubst95898992stappt @ Sigma @ U @ Eta2 ) ) ) ).
% usubstappf.simps(2)
thf(fact_233_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_234_pred__subset__eq,axiom,
! [A: $tType,R4: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ R4 )
@ ^ [X: A] : ( member @ A @ X @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R4 @ S ) ) ).
% pred_subset_eq
thf(fact_235_Collect__empty__eq__bot,axiom,
! [A: $tType,P3: A > $o] :
( ( ( collect @ A @ P3 )
= ( bot_bot @ ( set @ A ) ) )
= ( P3
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_236_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A5: set @ A] :
( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_237_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc2004651681e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_238_usubstappt_Osimps_I4_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,F2: char,Theta: trm] :
( ( uSubst95898992stappt @ Sigma @ U @ ( func @ F2 @ Theta ) )
= ( case_option @ ( option @ trm ) @ trm @ ( none @ trm )
@ ^ [Sigma_theta: trm] :
( case_option @ ( option @ trm ) @ trm @ ( some @ trm @ ( func @ F2 @ Sigma_theta ) )
@ ^ [R: trm] :
( if @ ( option @ trm )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R ) @ U )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898992stappt @ ( uSubst969145931substt @ Sigma_theta ) @ ( bot_bot @ ( set @ variable ) ) @ R )
@ ( none @ trm ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F4: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F4 ) )
@ Sigma
@ F2 ) )
@ ( uSubst95898992stappt @ Sigma @ U @ Theta ) ) ) ).
% usubstappt.simps(4)
thf(fact_239_trm_Oinject_I4_J,axiom,
! [X41: char,X42: trm,Y41: char,Y42: trm] :
( ( ( func @ X41 @ X42 )
= ( func @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% trm.inject(4)
thf(fact_240_usubstappt__func2,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),F2: char,R2: trm,U: set @ variable,Theta: trm] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F4: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F4 ) )
@ Sigma
@ F2 )
= ( some @ trm @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R2 ) @ U )
!= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( func @ F2 @ Theta ) )
= ( none @ trm ) ) ) ) ).
% usubstappt_func2
thf(fact_241_usubstappt__func,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),F2: char,R2: trm,U: set @ variable,Theta: trm,Sigma_theta2: trm] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F4: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F4 ) )
@ Sigma
@ F2 )
= ( some @ trm @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R2 ) @ U )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
= ( some @ trm @ Sigma_theta2 ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( func @ F2 @ Theta ) )
= ( uSubst95898992stappt @ ( uSubst969145931substt @ Sigma_theta2 ) @ ( bot_bot @ ( set @ variable ) ) @ R2 ) ) ) ) ) ).
% usubstappt_func
thf(fact_242_usubstappt__func__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,F2: char,Theta: trm] :
( ( ( uSubst95898992stappt @ Sigma @ U @ ( func @ F2 @ Theta ) )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
& ( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F4: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F4 ) )
@ Sigma
@ F2 )
= ( none @ trm ) )
| ? [R3: trm] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F4: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F4 ) )
@ Sigma
@ F2 )
= ( some @ trm @ R3 ) )
& ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R3 ) @ U )
= ( bot_bot @ ( set @ variable ) ) ) ) ) ) ) ).
% usubstappt_func_conv
thf(fact_243_usappconst__simp,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),F2: char,R2: trm,U: set @ variable] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F0: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F0 ) )
@ Sigma
@ F2 )
= ( some @ trm @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R2 ) @ U )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst1138577137pconst @ Sigma @ U @ F2 )
= ( some @ trm @ R2 ) ) ) ) ).
% usappconst_simp
thf(fact_244_usappconst__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,F2: char] :
( ( ( uSubst1138577137pconst @ Sigma @ U @ F2 )
!= ( none @ trm ) )
=> ( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F0: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F0 ) )
@ Sigma
@ F2 )
= ( none @ trm ) )
| ? [R3: trm] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F0: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F0 ) )
@ Sigma
@ F2 )
= ( some @ trm @ R3 ) )
& ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R3 ) @ U )
= ( bot_bot @ ( set @ variable ) ) ) ) ) ) ).
% usappconst_conv
thf(fact_245_usappconst__def,axiom,
( uSubst1138577137pconst
= ( ^ [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,F: char] :
( case_option @ ( option @ trm ) @ trm @ ( some @ trm @ ( const @ F ) )
@ ^ [R: trm] :
( if @ ( option @ trm )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R ) @ U2 )
= ( bot_bot @ ( set @ variable ) ) )
@ ( some @ trm @ R )
@ ( none @ trm ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F0: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F0 ) )
@ Sigma2
@ F ) ) ) ) ).
% usappconst_def
thf(fact_246_usubstappt__const,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),F2: char,R2: trm,U: set @ variable] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F0: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F0 ) )
@ Sigma
@ F2 )
= ( some @ trm @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R2 ) @ U )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( const @ F2 ) )
= ( some @ trm @ R2 ) ) ) ) ).
% usubstappt_const
thf(fact_247_trm_Oinject_I3_J,axiom,
! [X32: char,Y32: char] :
( ( ( const @ X32 )
= ( const @ Y32 ) )
= ( X32 = Y32 ) ) ).
% trm.inject(3)
thf(fact_248_usubstappt_Osimps_I3_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,F2: char] :
( ( uSubst95898992stappt @ Sigma @ U @ ( const @ F2 ) )
= ( uSubst1138577137pconst @ Sigma @ U @ F2 ) ) ).
% usubstappt.simps(3)
thf(fact_249_trm_Odistinct_I23_J,axiom,
! [X32: char,X41: char,X42: trm] :
( ( const @ X32 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(23)
thf(fact_250_usubstappt__const__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,F2: char] :
( ( ( uSubst95898992stappt @ Sigma @ U @ ( const @ F2 ) )
!= ( none @ trm ) )
=> ( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F0: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F0 ) )
@ Sigma
@ F2 )
= ( none @ trm ) )
| ? [R3: trm] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F0: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F0 ) )
@ Sigma
@ F2 )
= ( some @ trm @ R3 ) )
& ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R3 ) @ U )
= ( bot_bot @ ( set @ variable ) ) ) ) ) ) ).
% usubstappt_const_conv
thf(fact_251_subset__Collect__iff,axiom,
! [A: $tType,B3: set @ A,A4: set @ A,P3: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A4 )
& ( P3 @ X ) ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ B3 )
=> ( P3 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_252_subset__CollectI,axiom,
! [A: $tType,B3: set @ A,A4: set @ A,Q: A > $o,P3: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B3 @ A4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B3 )
=> ( ( Q @ X5 )
=> ( P3 @ X5 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ B3 )
& ( Q @ X ) ) )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A4 )
& ( P3 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_253_conj__subset__def,axiom,
! [A: $tType,A4: set @ A,P3: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A4
@ ( collect @ A
@ ^ [X: A] :
( ( P3 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P3 ) )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_254_Diamondo_Ocases,axiom,
! [X3: product_prod @ ( option @ game ) @ ( option @ fml )] :
( ! [Alpha: game,Phi2: fml] :
( X3
!= ( product_Pair @ ( option @ game ) @ ( option @ fml ) @ ( some @ game @ Alpha ) @ ( some @ fml @ Phi2 ) ) )
=> ( ! [Phi2: option @ fml] :
( X3
!= ( product_Pair @ ( option @ game ) @ ( option @ fml ) @ ( none @ game ) @ Phi2 ) )
=> ~ ! [V6: game] :
( X3
!= ( product_Pair @ ( option @ game ) @ ( option @ fml ) @ ( some @ game @ V6 ) @ ( none @ fml ) ) ) ) ) ).
% Diamondo.cases
% Type constructors (28)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A7: $tType,A8: $tType] :
( ( bounded_lattice @ A8 )
=> ( bounded_lattice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( bounded_lattice @ A8 )
=> ( bounded_lattice_bot @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_inf @ A8 )
=> ( semilattice_inf @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( order_bot @ A8 )
=> ( order_bot @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A7: $tType,A8: $tType] :
( ( lattice @ A8 )
=> ( lattice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A7: $tType,A8: $tType] :
( ( bot @ A8 )
=> ( bot @ ( A7 > A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_3,axiom,
! [A7: $tType] : ( bounded_lattice_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_4,axiom,
! [A7: $tType] : ( semilattice_inf @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_5,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_6,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_7,axiom,
! [A7: $tType] : ( lattice @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_10,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_11,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_12,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_13,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_15,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_16,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_17,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_18,axiom,
bot @ $o ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X3: A,Y3: A] :
( ( if @ A @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y3: A] :
( ( if @ A @ $true @ X3 @ Y3 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ( uSubst95898978stappf @ sigma @ va @ ( pred @ p @ theta ) )
!= ( none @ fml ) )
| ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p )
= ( none @ fml ) ) ) ).
%------------------------------------------------------------------------------