TPTP Problem File: ITP202^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP202^2 : TPTP v9.0.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer USubst problem prob_411__6333658_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_411__6333658_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 337 ( 149 unt; 61 typ; 0 def)
% Number of atoms : 695 ( 495 equ; 0 cnn)
% Maximal formula atoms : 31 ( 2 avg)
% Number of connectives : 7368 ( 199 ~; 10 |; 25 &;6770 @)
% ( 0 <=>; 364 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 1260 (1260 >; 0 *; 0 +; 0 <<)
% Number of symbols : 57 ( 54 usr; 7 con; 0-8 aty)
% Number of variables : 1395 ( 225 ^;1126 !; 14 ?;1395 :)
% ( 30 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:19:27.220
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_t_Denotational__Semantics_Ointerp,type,
denotational_interp: $tType ).
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_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
% Explicit typings (51)
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_Lattices_Olattice,type,
lattice:
!>[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_Ocombine__options,type,
combine_options:
!>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ 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_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_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
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_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
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_ODifferential,type,
differential: trm > trm ).
thf(sy_c_Syntax_Otrm_OFunc,type,
func: char > trm > trm ).
thf(sy_c_Syntax_Otrm_ONumber,type,
number: real > trm ).
thf(sy_c_Syntax_Otrm_OPlus,type,
plus: trm > trm > trm ).
thf(sy_c_Syntax_Otrm_OTimes,type,
times: trm > trm > trm ).
thf(sy_c_Syntax_Otrm_OVar,type,
var: variable > trm ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_ODifferentialo,type,
uSubst259074819ntialo: ( option @ trm ) > ( option @ trm ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OGeqo,type,
uSubst1556497037e_Geqo: ( option @ trm ) > ( option @ trm ) > ( option @ fml ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OGeqo__rel,type,
uSubst864323244qo_rel: ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) > ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) > $o ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OPluso,type,
uSubst1112714340_Pluso: ( option @ trm ) > ( option @ trm ) > ( option @ trm ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OPluso__rel,type,
uSubst270600597so_rel: ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) > ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) > $o ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OTimeso,type,
uSubst277968634Timeso: ( option @ trm ) > ( option @ trm ) > ( option @ trm ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OTimeso__rel,type,
uSubst1377811071so_rel: ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) > ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) > $o ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_Odot,type,
uSubst_Mirabelle_dot: trm ).
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_USubst__Mirabelle__nnnzepxswx_Ousubstappt__rel,type,
uSubst2096773001pt_rel: ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) > ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) > $o ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_U,type,
u: set @ variable ).
thf(sy_v_V,type,
v: 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_f____,type,
f: char ).
% Relevant facts (255)
thf(fact_0_f2,axiom,
! [Z: option @ trm,F: trm > ( option @ trm ),Za: option @ trm] :
( ( ( Za
= ( none @ trm ) )
=> ( ( case_option @ ( option @ trm ) @ trm @ Z @ F @ Za )
= Z ) )
& ( ( Za
!= ( none @ trm ) )
=> ( ( case_option @ ( option @ trm ) @ trm @ Z @ F @ Za )
= ( F @ ( the @ trm @ Za ) ) ) ) ) ).
% f2
thf(fact_1_Const_Oprems_I1_J,axiom,
( ( uSubst95898992stappt @ sigma @ u @ ( const @ f ) )
!= ( none @ trm ) ) ).
% Const.prems(1)
thf(fact_2_Const_Oprems_I2_J,axiom,
( ( uSubst95898992stappt @ sigma @ v @ ( const @ f ) )
!= ( none @ trm ) ) ).
% Const.prems(2)
thf(fact_3_f3,axiom,
( ( ( 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
@ f )
!= ( none @ trm ) )
=> ( ( inf_inf @ ( set @ variable )
@ ( static_FVT
@ ( the @ 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
@ f ) ) )
@ u )
= ( bot_bot @ ( set @ variable ) ) ) ) ).
% f3
thf(fact_4__092_060open_062SConst_A_092_060sigma_062_Af_A_092_060noteq_062_Aundeft_A_092_060longrightarrow_062_AFVT_A_Ithe_A_ISConst_A_092_060sigma_062_Af_J_J_A_092_060inter_062_AV_A_061_A_123_125_092_060close_062,axiom,
( ( ( 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
@ f )
!= ( none @ trm ) )
=> ( ( inf_inf @ ( set @ variable )
@ ( static_FVT
@ ( the @ 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
@ f ) ) )
@ v )
= ( bot_bot @ ( set @ variable ) ) ) ) ).
% \<open>SConst \<sigma> f \<noteq> undeft \<longrightarrow> FVT (the (SConst \<sigma> f)) \<inter> V = {}\<close>
thf(fact_5_usappconst__def,axiom,
( uSubst1138577137pconst
= ( ^ [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,F2: char] :
( case_option @ ( option @ trm ) @ trm @ ( some @ trm @ ( const @ F2 ) )
@ ^ [R: trm] :
( if @ ( option @ trm )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R ) @ U )
= ( 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 ) )
@ Sigma
@ F2 ) ) ) ) ).
% usappconst_def
thf(fact_6_f1,axiom,
( ( uSubst95898992stappt @ sigma @ u @ ( const @ f ) )
= ( case_option @ ( option @ trm ) @ trm @ ( some @ trm @ ( const @ f ) )
@ ^ [T2: trm] :
( if @ ( option @ trm )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ T2 ) @ u )
= ( bot_bot @ ( set @ variable ) ) )
@ ( some @ trm @ T2 )
@ ( 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
@ f ) ) ) ).
% f1
thf(fact_7_usubstappt_Osimps_I3_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,F3: char] :
( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( const @ F3 ) )
= ( uSubst1138577137pconst @ Sigma2 @ U2 @ F3 ) ) ).
% usubstappt.simps(3)
thf(fact_8_undeft__None,axiom,
( ( none @ trm )
= ( none @ trm ) ) ).
% undeft_None
thf(fact_9__092_060open_062SConst_A_092_060sigma_062_Af_A_092_060noteq_062_Aundeft_A_092_060longrightarrow_062_A_Iif_AFVT_A_Ithe_A_ISConst_A_092_060sigma_062_Af_J_J_A_092_060inter_062_AV_A_061_A_123_125_Athen_AAterm_A_Ithe_A_ISConst_A_092_060sigma_062_Af_J_J_Aelse_Aundeft_J_A_061_Ausappconst_A_092_060sigma_062_AV_Af_092_060close_062,axiom,
( ( ( 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
@ f )
!= ( none @ trm ) )
=> ( ( ( ( inf_inf @ ( set @ variable )
@ ( static_FVT
@ ( the @ 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
@ f ) ) )
@ v )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( some @ trm
@ ( the @ 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
@ f ) ) )
= ( uSubst1138577137pconst @ sigma @ v @ f ) ) )
& ( ( ( inf_inf @ ( set @ variable )
@ ( static_FVT
@ ( the @ 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
@ f ) ) )
@ v )
!= ( bot_bot @ ( set @ variable ) ) )
=> ( ( none @ trm )
= ( uSubst1138577137pconst @ sigma @ v @ f ) ) ) ) ) ).
% \<open>SConst \<sigma> f \<noteq> undeft \<longrightarrow> (if FVT (the (SConst \<sigma> f)) \<inter> V = {} then Aterm (the (SConst \<sigma> f)) else undeft) = usappconst \<sigma> V f\<close>
thf(fact_10__092_060open_062SConst_A_092_060sigma_062_Af_A_092_060noteq_062_Aundeft_A_092_060longrightarrow_062_A_Iif_AFVT_A_Ithe_A_ISConst_A_092_060sigma_062_Af_J_J_A_092_060inter_062_AU_A_061_A_123_125_Athen_AAterm_A_Ithe_A_ISConst_A_092_060sigma_062_Af_J_J_Aelse_Aundeft_J_A_061_Ausappconst_A_092_060sigma_062_AU_Af_092_060close_062,axiom,
( ( ( 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
@ f )
!= ( none @ trm ) )
=> ( ( ( ( inf_inf @ ( set @ variable )
@ ( static_FVT
@ ( the @ 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
@ f ) ) )
@ u )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( some @ trm
@ ( the @ 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
@ f ) ) )
= ( uSubst1138577137pconst @ sigma @ u @ f ) ) )
& ( ( ( inf_inf @ ( set @ variable )
@ ( static_FVT
@ ( the @ 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
@ f ) ) )
@ u )
!= ( bot_bot @ ( set @ variable ) ) )
=> ( ( none @ trm )
= ( uSubst1138577137pconst @ sigma @ u @ f ) ) ) ) ) ).
% \<open>SConst \<sigma> f \<noteq> undeft \<longrightarrow> (if FVT (the (SConst \<sigma> f)) \<inter> U = {} then Aterm (the (SConst \<sigma> f)) else undeft) = usappconst \<sigma> U f\<close>
thf(fact_11_trm_Oinject_I3_J,axiom,
! [X3: char,Y3: char] :
( ( ( const @ X3 )
= ( const @ Y3 ) )
= ( X3 = Y3 ) ) ).
% trm.inject(3)
thf(fact_12_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F2: B > C > D > A,X: product_prod @ B @ C,Y: D] :
( product_case_prod @ B @ C @ A
@ ^ [L: B,R: C] : ( F2 @ L @ R @ Y )
@ X ) ) ) ).
% case_prod_app
thf(fact_13_usubstappt__const,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),F3: char,R2: trm,U2: 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 ) )
@ Sigma2
@ F3 )
= ( some @ trm @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R2 ) @ U2 )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( const @ F3 ) )
= ( some @ trm @ R2 ) ) ) ) ).
% usubstappt_const
thf(fact_14_usubstappt__const__conv,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,F3: char] :
( ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( const @ F3 ) )
!= ( 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
@ F3 )
= ( 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 ) )
@ Sigma2
@ F3 )
= ( some @ trm @ R3 ) )
& ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R3 ) @ U2 )
= ( bot_bot @ ( set @ variable ) ) ) ) ) ) ).
% usubstappt_const_conv
thf(fact_15_usappconst__simp,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),F3: char,R2: trm,U2: 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 ) )
@ Sigma2
@ F3 )
= ( some @ trm @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R2 ) @ U2 )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst1138577137pconst @ Sigma2 @ U2 @ F3 )
= ( some @ trm @ R2 ) ) ) ) ).
% usappconst_simp
thf(fact_16_usappconst__conv,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,F3: char] :
( ( ( uSubst1138577137pconst @ Sigma2 @ U2 @ F3 )
!= ( 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
@ F3 )
= ( 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 ) )
@ Sigma2
@ F3 )
= ( some @ trm @ R3 ) )
& ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R3 ) @ U2 )
= ( bot_bot @ ( set @ variable ) ) ) ) ) ) ).
% usappconst_conv
thf(fact_17_ODEo_Oinduct,axiom,
! [P: char > ( option @ trm ) > $o,A0: char,A1: option @ trm] :
( ! [X2: char,Theta: trm] : ( P @ X2 @ ( some @ trm @ Theta ) )
=> ( ! [X2: char] : ( P @ X2 @ ( none @ trm ) )
=> ( P @ A0 @ A1 ) ) ) ).
% ODEo.induct
thf(fact_18_undeft__equiv,axiom,
! [Theta2: option @ trm] :
( ( Theta2
!= ( none @ trm ) )
= ( ? [T2: trm] :
( Theta2
= ( some @ trm @ T2 ) ) ) ) ).
% undeft_equiv
thf(fact_19_Timeso_Oinduct,axiom,
! [P: ( option @ trm ) > ( option @ trm ) > $o,A0: option @ trm,A1: option @ trm] :
( ! [Theta: trm,Eta: trm] : ( P @ ( some @ trm @ Theta ) @ ( some @ trm @ Eta ) )
=> ( ! [X_1: option @ trm] : ( P @ ( none @ trm ) @ X_1 )
=> ( ! [V: trm] : ( P @ ( some @ trm @ V ) @ ( none @ trm ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% Timeso.induct
thf(fact_20_Assigno_Oinduct,axiom,
! [P: variable > ( option @ trm ) > $o,A0: variable,A1: option @ trm] :
( ! [X2: variable,Theta: trm] : ( P @ X2 @ ( some @ trm @ Theta ) )
=> ( ! [X2: variable] : ( P @ X2 @ ( none @ trm ) )
=> ( P @ A0 @ A1 ) ) ) ).
% Assigno.induct
thf(fact_21_Aterm__Some,axiom,
( ( some @ trm )
= ( some @ trm ) ) ).
% Aterm_Some
thf(fact_22_Differentialo_Oinduct,axiom,
! [P: ( option @ trm ) > $o,A0: option @ trm] :
( ! [Theta: trm] : ( P @ ( some @ trm @ Theta ) )
=> ( ( P @ ( none @ trm ) )
=> ( P @ A0 ) ) ) ).
% Differentialo.induct
thf(fact_23_Differentialo_Ocases,axiom,
! [X4: option @ trm] :
( ! [Theta: trm] :
( X4
!= ( some @ trm @ Theta ) )
=> ( X4
= ( none @ trm ) ) ) ).
% Differentialo.cases
thf(fact_24_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_25_not__Some__eq,axiom,
! [A: $tType,X4: option @ A] :
( ( ! [Y: A] :
( X4
!= ( some @ A @ Y ) ) )
= ( X4
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_26_not__None__eq,axiom,
! [A: $tType,X4: option @ A] :
( ( X4
!= ( none @ A ) )
= ( ? [Y: A] :
( X4
= ( some @ A @ Y ) ) ) ) ).
% not_None_eq
thf(fact_27_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X4: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X4 )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_28_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X4: A] :
( ( inf_inf @ A @ X4 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_29_usubstappt__func2,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),F3: char,R2: trm,U2: set @ variable,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 ) )
@ Sigma2
@ F3 )
= ( some @ trm @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R2 ) @ U2 )
!= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( func @ F3 @ Theta2 ) )
= ( none @ trm ) ) ) ) ).
% usubstappt_func2
thf(fact_30_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P @ F1 ) )
| ( ( Option
= ( some @ A @ ( the @ A @ Option ) ) )
& ~ ( P @ ( F22 @ ( the @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_31_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P @ F1 ) )
& ( ( Option
= ( some @ A @ ( the @ A @ Option ) ) )
=> ( P @ ( F22 @ ( the @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_32_usubstappt__func__conv,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,F3: char,Theta2: trm] :
( ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( func @ F3 @ Theta2 ) )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 )
!= ( 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 ) )
@ Sigma2
@ F3 )
= ( 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 ) )
@ Sigma2
@ F3 )
= ( some @ trm @ R3 ) )
& ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R3 ) @ U2 )
= ( bot_bot @ ( set @ variable ) ) ) ) ) ) ) ).
% usubstappt_func_conv
thf(fact_33_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_34_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,Y2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X4 @ Y2 ) @ Y2 )
= ( inf_inf @ A @ X4 @ Y2 ) ) ) ).
% inf_right_idem
thf(fact_35_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.right_idem
thf(fact_36_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,Y2: A] :
( ( inf_inf @ A @ X4 @ ( inf_inf @ A @ X4 @ Y2 ) )
= ( inf_inf @ A @ X4 @ Y2 ) ) ) ).
% inf_left_idem
thf(fact_37_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.left_idem
thf(fact_38_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A] :
( ( inf_inf @ A @ X4 @ X4 )
= X4 ) ) ).
% inf_idem
thf(fact_39_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A] :
( ( inf_inf @ A @ A2 @ A2 )
= A2 ) ) ).
% inf.idem
thf(fact_40_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B,X: A] : ( inf_inf @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% inf_apply
thf(fact_41_option_Oinject,axiom,
! [A: $tType,X22: A,Y22: A] :
( ( ( some @ A @ X22 )
= ( some @ A @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_42_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_43_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_44_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_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G2: A > B] :
( ! [X2: A] :
( ( F3 @ X2 )
= ( G2 @ X2 ) )
=> ( F3 = G2 ) ) ).
% ext
thf(fact_49_trm_Odistinct_I23_J,axiom,
! [X3: char,X41: char,X42: trm] :
( ( const @ X3 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(23)
thf(fact_50_case__optionE,axiom,
! [A: $tType,P: $o,Q: A > $o,X4: option @ A] :
( ( case_option @ $o @ A @ P @ Q @ X4 )
=> ( ( ( X4
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y4: A] :
( ( X4
= ( some @ A @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_51_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ X4 @ ( inf_inf @ A @ Y2 @ Z2 ) )
= ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X4 @ Z2 ) ) ) ) ).
% inf_left_commute
thf(fact_52_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C2 ) )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.left_commute
thf(fact_53_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X: A,Y: A] : ( inf_inf @ A @ Y @ X ) ) ) ) ).
% inf_commute
thf(fact_54_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A4: A,B3: A] : ( inf_inf @ A @ B3 @ A4 ) ) ) ) ).
% inf.commute
thf(fact_55_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X4: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X4 @ Y2 ) @ Z2 )
= ( inf_inf @ A @ X4 @ ( inf_inf @ A @ Y2 @ Z2 ) ) ) ) ).
% inf_assoc
thf(fact_56_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.assoc
thf(fact_57_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B4: A,K: A,B2: A,A2: A] :
( ( B4
= ( inf_inf @ A @ K @ B2 ) )
=> ( ( inf_inf @ A @ A2 @ B4 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_58_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,K: A,A2: A,B2: A] :
( ( A3
= ( inf_inf @ A @ K @ A2 ) )
=> ( ( inf_inf @ A @ A3 @ B2 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_59_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B,X: A] : ( inf_inf @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% inf_fun_def
thf(fact_60_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_61_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X4: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X4 @ Y2 ) @ Z2 )
= ( inf_inf @ A @ X4 @ ( inf_inf @ A @ Y2 @ Z2 ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_62_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X4: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ X4 @ ( inf_inf @ A @ Y2 @ Z2 ) )
= ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X4 @ Z2 ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_63_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X4: A,Y2: A] :
( ( inf_inf @ A @ X4 @ ( inf_inf @ A @ X4 @ Y2 ) )
= ( inf_inf @ A @ X4 @ Y2 ) ) ) ).
% inf_sup_aci(4)
thf(fact_64_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_65_option_Odistinct_I1_J,axiom,
! [A: $tType,X22: A] :
( ( none @ A )
!= ( some @ A @ X22 ) ) ).
% option.distinct(1)
thf(fact_66_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X22: A] :
( ( Option
= ( some @ A @ X22 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_67_option_Oexhaust,axiom,
! [A: $tType,Y2: option @ A] :
( ( Y2
!= ( none @ A ) )
=> ~ ! [X23: A] :
( Y2
!= ( some @ A @ X23 ) ) ) ).
% option.exhaust
thf(fact_68_option_Oinducts,axiom,
! [A: $tType,P: ( option @ A ) > $o,Option: option @ A] :
( ( P @ ( none @ A ) )
=> ( ! [X2: A] : ( P @ ( some @ A @ X2 ) )
=> ( P @ Option ) ) ) ).
% option.inducts
thf(fact_69_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
? [X5: option @ A] : ( P2 @ X5 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
| ? [X: A] : ( P3 @ ( some @ A @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_70_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
! [X5: option @ A] : ( P2 @ X5 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
& ! [X: A] : ( P3 @ ( some @ A @ X ) ) ) ) ) ).
% split_option_all
thf(fact_71_combine__options__cases,axiom,
! [A: $tType,B: $tType,X4: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y2: option @ B] :
( ( ( X4
= ( none @ A ) )
=> ( P @ X4 @ Y2 ) )
=> ( ( ( Y2
= ( none @ B ) )
=> ( P @ X4 @ Y2 ) )
=> ( ! [A5: A,B5: B] :
( ( X4
= ( some @ A @ A5 ) )
=> ( ( Y2
= ( some @ B @ B5 ) )
=> ( P @ X4 @ Y2 ) ) )
=> ( P @ X4 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_72_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_73_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_74_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_75_option_Osel,axiom,
! [A: $tType,X22: A] :
( ( the @ A @ ( some @ A @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_76_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_77_usubstappt__func,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),F3: char,R2: trm,U2: set @ variable,Theta2: trm,Sigma_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 ) )
@ Sigma2
@ F3 )
= ( some @ trm @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R2 ) @ U2 )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 )
= ( some @ trm @ Sigma_theta ) )
=> ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( func @ F3 @ Theta2 ) )
= ( uSubst95898992stappt @ ( uSubst969145931substt @ Sigma_theta ) @ ( bot_bot @ ( set @ variable ) ) @ R2 ) ) ) ) ) ).
% usubstappt_func
thf(fact_78_usubstappt_Osimps_I4_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,F3: char,Theta2: trm] :
( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( func @ F3 @ Theta2 ) )
= ( case_option @ ( option @ trm ) @ trm @ ( none @ trm )
@ ^ [Sigma_theta2: trm] :
( case_option @ ( option @ trm ) @ trm @ ( some @ trm @ ( func @ F3 @ Sigma_theta2 ) )
@ ^ [R: trm] :
( if @ ( option @ trm )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R ) @ U2 )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898992stappt @ ( uSubst969145931substt @ Sigma_theta2 ) @ ( 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 ) )
@ Sigma2
@ F3 ) )
@ ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 ) ) ) ).
% usubstappt.simps(4)
thf(fact_79_Int__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
= ( ( member @ A @ C2 @ A3 )
& ( member @ A @ C2 @ B4 ) ) ) ).
% Int_iff
thf(fact_80_IntI,axiom,
! [A: $tType,C2: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) ) ) ) ).
% IntI
thf(fact_81_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X: A] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_82_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_83_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X: A] :
~ ( member @ A @ X @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_84_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_85_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_86_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_87_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_88_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_89_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_90_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X: A] : ( member @ A @ X @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_91_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_92_IntE,axiom,
! [A: $tType,C2: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
=> ~ ( ( member @ A @ C2 @ A3 )
=> ~ ( member @ A @ C2 @ B4 ) ) ) ).
% IntE
thf(fact_93_IntD1,axiom,
! [A: $tType,C2: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
=> ( member @ A @ C2 @ A3 ) ) ).
% IntD1
thf(fact_94_IntD2,axiom,
! [A: $tType,C2: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
=> ( member @ A @ C2 @ B4 ) ) ).
% IntD2
thf(fact_95_Int__assoc,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) @ C3 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% Int_assoc
thf(fact_96_Int__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_97_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% Int_commute
thf(fact_98_Int__left__absorb,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ A3 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A3 @ B4 ) ) ).
% Int_left_absorb
thf(fact_99_Int__left__commute,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( inf_inf @ ( set @ A ) @ B4 @ C3 ) )
= ( inf_inf @ ( set @ A ) @ B4 @ ( inf_inf @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_100_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X: A] : $false ) ) ).
% empty_def
thf(fact_101_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_102_Int__Collect,axiom,
! [A: $tType,X4: A,A3: set @ A,P: A > $o] :
( ( member @ A @ X4 @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X4 @ A3 )
& ( P @ X4 ) ) ) ).
% Int_Collect
thf(fact_103_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A6 )
& ( member @ A @ X @ B6 ) ) ) ) ) ).
% Int_def
thf(fact_104_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A6 )
@ ^ [X: A] : ( member @ A @ X @ B6 ) ) ) ) ) ).
% inf_set_def
thf(fact_105_Int__emptyI,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ~ ( member @ A @ X2 @ B4 ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_106_disjoint__iff,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ~ ( member @ A @ X @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_107_Int__empty__left,axiom,
! [A: $tType,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_108_Int__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_109_disjoint__iff__not__equal,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ! [Y: A] :
( ( member @ A @ Y @ B4 )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_110_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > $o,X4: option @ A,R4: B > $o,F3: B,G2: A > B] :
( ( case_option @ $o @ A @ P @ Q @ X4 )
=> ( ( P
=> ( R4 @ F3 ) )
=> ( ! [Q2: A] :
( ( Q @ Q2 )
=> ( R4 @ ( G2 @ Q2 ) ) )
=> ( R4 @ ( case_option @ B @ A @ F3 @ G2 @ X4 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_111_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_112_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X1: A,X24: B] : ( H @ ( F3 @ X1 @ X24 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_113_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_114_Timeso_Osimps_I3_J,axiom,
! [V2: trm] :
( ( uSubst277968634Timeso @ ( some @ trm @ V2 ) @ ( none @ trm ) )
= ( none @ trm ) ) ).
% Timeso.simps(3)
thf(fact_115_Pluso_Osimps_I3_J,axiom,
! [V2: trm] :
( ( uSubst1112714340_Pluso @ ( some @ trm @ V2 ) @ ( none @ trm ) )
= ( none @ trm ) ) ).
% Pluso.simps(3)
thf(fact_116_inf1I,axiom,
! [A: $tType,A3: A > $o,X4: A,B4: A > $o] :
( ( A3 @ X4 )
=> ( ( B4 @ X4 )
=> ( inf_inf @ ( A > $o ) @ A3 @ B4 @ X4 ) ) ) ).
% inf1I
thf(fact_117_inf1E,axiom,
! [A: $tType,A3: A > $o,B4: A > $o,X4: A] :
( ( inf_inf @ ( A > $o ) @ A3 @ B4 @ X4 )
=> ~ ( ( A3 @ X4 )
=> ~ ( B4 @ X4 ) ) ) ).
% inf1E
thf(fact_118_inf1D1,axiom,
! [A: $tType,A3: A > $o,B4: A > $o,X4: A] :
( ( inf_inf @ ( A > $o ) @ A3 @ B4 @ X4 )
=> ( A3 @ X4 ) ) ).
% inf1D1
thf(fact_119_inf1D2,axiom,
! [A: $tType,A3: A > $o,B4: A > $o,X4: A] :
( ( inf_inf @ ( A > $o ) @ A3 @ B4 @ X4 )
=> ( B4 @ X4 ) ) ).
% inf1D2
thf(fact_120_Pluso_Osimps_I2_J,axiom,
! [Eta2: option @ trm] :
( ( uSubst1112714340_Pluso @ ( none @ trm ) @ Eta2 )
= ( none @ trm ) ) ).
% Pluso.simps(2)
thf(fact_121_Timeso_Osimps_I2_J,axiom,
! [Eta2: option @ trm] :
( ( uSubst277968634Timeso @ ( none @ trm ) @ Eta2 )
= ( none @ trm ) ) ).
% Timeso.simps(2)
thf(fact_122_Pluso__undef,axiom,
! [Theta2: option @ trm,Eta2: option @ trm] :
( ( ( uSubst1112714340_Pluso @ Theta2 @ Eta2 )
= ( none @ trm ) )
= ( ( Theta2
= ( none @ trm ) )
| ( Eta2
= ( none @ trm ) ) ) ) ).
% Pluso_undef
thf(fact_123_Timeso__undef,axiom,
! [Theta2: option @ trm,Eta2: option @ trm] :
( ( ( uSubst277968634Timeso @ Theta2 @ Eta2 )
= ( none @ trm ) )
= ( ( Theta2
= ( none @ trm ) )
| ( Eta2
= ( none @ trm ) ) ) ) ).
% Timeso_undef
thf(fact_124_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_125_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_126_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_127_Pluso_Oelims,axiom,
! [X4: option @ trm,Xa: option @ trm,Y2: option @ trm] :
( ( ( uSubst1112714340_Pluso @ X4 @ Xa )
= Y2 )
=> ( ! [Theta: trm] :
( ( X4
= ( some @ trm @ Theta ) )
=> ! [Eta: trm] :
( ( Xa
= ( some @ trm @ Eta ) )
=> ( Y2
!= ( some @ trm @ ( plus @ Theta @ Eta ) ) ) ) )
=> ( ( ( X4
= ( none @ trm ) )
=> ( Y2
!= ( none @ trm ) ) )
=> ~ ( ? [V: trm] :
( X4
= ( some @ trm @ V ) )
=> ( ( Xa
= ( none @ trm ) )
=> ( Y2
!= ( none @ trm ) ) ) ) ) ) ) ).
% Pluso.elims
thf(fact_128_Timeso_Oelims,axiom,
! [X4: option @ trm,Xa: option @ trm,Y2: option @ trm] :
( ( ( uSubst277968634Timeso @ X4 @ Xa )
= Y2 )
=> ( ! [Theta: trm] :
( ( X4
= ( some @ trm @ Theta ) )
=> ! [Eta: trm] :
( ( Xa
= ( some @ trm @ Eta ) )
=> ( Y2
!= ( some @ trm @ ( times @ Theta @ Eta ) ) ) ) )
=> ( ( ( X4
= ( none @ trm ) )
=> ( Y2
!= ( none @ trm ) ) )
=> ~ ( ? [V: trm] :
( X4
= ( some @ trm @ V ) )
=> ( ( Xa
= ( none @ trm ) )
=> ( Y2
!= ( none @ trm ) ) ) ) ) ) ) ).
% Timeso.elims
thf(fact_129_trm_Oinject_I5_J,axiom,
! [X51: trm,X52: trm,Y51: trm,Y52: trm] :
( ( ( plus @ X51 @ X52 )
= ( plus @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% trm.inject(5)
thf(fact_130_trm_Oinject_I6_J,axiom,
! [X61: trm,X62: trm,Y61: trm,Y62: trm] :
( ( ( times @ X61 @ X62 )
= ( times @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% trm.inject(6)
thf(fact_131_trm_Odistinct_I37_J,axiom,
! [X51: trm,X52: trm,X61: trm,X62: trm] :
( ( plus @ X51 @ X52 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(37)
thf(fact_132_trm_Odistinct_I25_J,axiom,
! [X3: char,X51: trm,X52: trm] :
( ( const @ X3 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(25)
thf(fact_133_trm_Odistinct_I27_J,axiom,
! [X3: char,X61: trm,X62: trm] :
( ( const @ X3 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(27)
thf(fact_134_trm_Odistinct_I31_J,axiom,
! [X41: char,X42: trm,X51: trm,X52: trm] :
( ( func @ X41 @ X42 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(31)
thf(fact_135_trm_Odistinct_I33_J,axiom,
! [X41: char,X42: trm,X61: trm,X62: trm] :
( ( func @ X41 @ X42 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(33)
thf(fact_136_usubstappt__times__conv,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,Theta2: trm,Eta2: trm] :
( ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( times @ Theta2 @ Eta2 ) )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 )
!= ( none @ trm ) )
& ( ( uSubst95898992stappt @ Sigma2 @ U2 @ Eta2 )
!= ( none @ trm ) ) ) ) ).
% usubstappt_times_conv
thf(fact_137_usubstappt__plus__conv,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,Theta2: trm,Eta2: trm] :
( ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( plus @ Theta2 @ Eta2 ) )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 )
!= ( none @ trm ) )
& ( ( uSubst95898992stappt @ Sigma2 @ U2 @ Eta2 )
!= ( none @ trm ) ) ) ) ).
% usubstappt_plus_conv
thf(fact_138_Timeso_Osimps_I1_J,axiom,
! [Theta2: trm,Eta2: trm] :
( ( uSubst277968634Timeso @ ( some @ trm @ Theta2 ) @ ( some @ trm @ Eta2 ) )
= ( some @ trm @ ( times @ Theta2 @ Eta2 ) ) ) ).
% Timeso.simps(1)
thf(fact_139_Pluso_Osimps_I1_J,axiom,
! [Theta2: trm,Eta2: trm] :
( ( uSubst1112714340_Pluso @ ( some @ trm @ Theta2 ) @ ( some @ trm @ Eta2 ) )
= ( some @ trm @ ( plus @ Theta2 @ Eta2 ) ) ) ).
% Pluso.simps(1)
thf(fact_140_usubstappt_Osimps_I6_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,Theta2: trm,Eta2: trm] :
( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( times @ Theta2 @ Eta2 ) )
= ( uSubst277968634Timeso @ ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 ) @ ( uSubst95898992stappt @ Sigma2 @ U2 @ Eta2 ) ) ) ).
% usubstappt.simps(6)
thf(fact_141_usubstappt_Osimps_I5_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,Theta2: trm,Eta2: trm] :
( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( plus @ Theta2 @ Eta2 ) )
= ( uSubst1112714340_Pluso @ ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 ) @ ( uSubst95898992stappt @ Sigma2 @ U2 @ Eta2 ) ) ) ).
% usubstappt.simps(5)
thf(fact_142_usubstappt_Oelims,axiom,
! [X4: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Xa: set @ variable,Xb: trm,Y2: option @ trm] :
( ( ( uSubst95898992stappt @ X4 @ Xa @ Xb )
= Y2 )
=> ( ! [X2: variable] :
( ( Xb
= ( var @ X2 ) )
=> ( Y2
!= ( some @ trm @ ( var @ X2 ) ) ) )
=> ( ! [R3: real] :
( ( Xb
= ( number @ R3 ) )
=> ( Y2
!= ( some @ trm @ ( number @ R3 ) ) ) )
=> ( ! [F5: char] :
( ( Xb
= ( const @ F5 ) )
=> ( Y2
!= ( uSubst1138577137pconst @ X4 @ Xa @ F5 ) ) )
=> ( ! [F5: char,Theta: trm] :
( ( Xb
= ( func @ F5 @ Theta ) )
=> ( Y2
!= ( case_option @ ( option @ trm ) @ trm @ ( none @ trm )
@ ^ [Sigma_theta2: trm] :
( case_option @ ( option @ trm ) @ trm @ ( some @ trm @ ( func @ F5 @ Sigma_theta2 ) )
@ ^ [R: trm] :
( if @ ( option @ trm )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R ) @ Xa )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898992stappt @ ( uSubst969145931substt @ Sigma_theta2 ) @ ( 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 ) )
@ X4
@ F5 ) )
@ ( uSubst95898992stappt @ X4 @ Xa @ Theta ) ) ) )
=> ( ! [Theta: trm,Eta: trm] :
( ( Xb
= ( plus @ Theta @ Eta ) )
=> ( Y2
!= ( uSubst1112714340_Pluso @ ( uSubst95898992stappt @ X4 @ Xa @ Theta ) @ ( uSubst95898992stappt @ X4 @ Xa @ Eta ) ) ) )
=> ( ! [Theta: trm,Eta: trm] :
( ( Xb
= ( times @ Theta @ Eta ) )
=> ( Y2
!= ( uSubst277968634Timeso @ ( uSubst95898992stappt @ X4 @ Xa @ Theta ) @ ( uSubst95898992stappt @ X4 @ Xa @ Eta ) ) ) )
=> ~ ! [Theta: trm] :
( ( Xb
= ( differential @ Theta ) )
=> ( Y2
!= ( uSubst259074819ntialo
@ ( uSubst95898992stappt @ X4
@ ( collect @ variable
@ ^ [X: variable] : $true )
@ Theta ) ) ) ) ) ) ) ) ) ) ) ).
% usubstappt.elims
thf(fact_143_usubstappt__induct,axiom,
! [P: ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) > ( set @ variable ) > trm > $o,A0: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),A1: set @ variable,A22: trm] :
( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,X2: variable] : ( P @ Sigma3 @ U3 @ ( var @ X2 ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,R3: real] : ( P @ Sigma3 @ U3 @ ( number @ R3 ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,F5: char] : ( P @ Sigma3 @ U3 @ ( const @ F5 ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,F5: char,Theta: trm] :
( ( P @ Sigma3 @ U3 @ Theta )
=> ( ! [X25: trm] :
( ( ( uSubst95898992stappt @ Sigma3 @ U3 @ Theta )
= ( some @ trm @ X25 ) )
=> ! [X2a: 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 ) )
@ Sigma3
@ F5 )
= ( some @ trm @ X2a ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ X2a ) @ U3 )
= ( bot_bot @ ( set @ variable ) ) )
=> ( P @ ( uSubst969145931substt @ X25 ) @ ( bot_bot @ ( set @ variable ) ) @ X2a ) ) ) )
=> ( P @ Sigma3 @ U3 @ ( func @ F5 @ Theta ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta: trm] :
( ( P @ Sigma3 @ U3 @ Theta )
=> ! [Eta: trm] :
( ( P @ Sigma3 @ U3 @ Eta )
=> ( P @ Sigma3 @ U3 @ ( plus @ Theta @ Eta ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta: trm] :
( ( P @ Sigma3 @ U3 @ Theta )
=> ! [Eta: trm] :
( ( P @ Sigma3 @ U3 @ Eta )
=> ( P @ Sigma3 @ U3 @ ( times @ Theta @ Eta ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta: trm] :
( ( P @ Sigma3
@ ( collect @ variable
@ ^ [X: variable] : $true )
@ Theta )
=> ( P @ Sigma3 @ U3 @ ( differential @ Theta ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ) ) ) ).
% usubstappt_induct
thf(fact_144_Geqo_Osimps_I3_J,axiom,
! [V2: trm] :
( ( uSubst1556497037e_Geqo @ ( some @ trm @ V2 ) @ ( none @ trm ) )
= ( none @ fml ) ) ).
% Geqo.simps(3)
thf(fact_145_combine__options__def,axiom,
! [A: $tType] :
( ( combine_options @ A )
= ( ^ [F2: A > A > A,X: option @ A,Y: option @ A] :
( case_option @ ( option @ A ) @ A @ Y
@ ^ [Z3: A] :
( case_option @ ( option @ A ) @ A @ ( some @ A @ Z3 )
@ ^ [Aa: A] : ( some @ A @ ( F2 @ Z3 @ Aa ) )
@ Y )
@ X ) ) ) ).
% combine_options_def
thf(fact_146_trm_Oinject_I2_J,axiom,
! [X22: real,Y22: real] :
( ( ( number @ X22 )
= ( number @ Y22 ) )
= ( X22 = Y22 ) ) ).
% trm.inject(2)
thf(fact_147_trm_Oinject_I1_J,axiom,
! [X12: variable,Y1: variable] :
( ( ( var @ X12 )
= ( var @ Y1 ) )
= ( X12 = Y1 ) ) ).
% trm.inject(1)
thf(fact_148_trm_Oinject_I7_J,axiom,
! [X7: trm,Y7: trm] :
( ( ( differential @ X7 )
= ( differential @ Y7 ) )
= ( X7 = Y7 ) ) ).
% trm.inject(7)
thf(fact_149_combine__options__simps_I2_J,axiom,
! [A: $tType,F3: A > A > A,X4: option @ A] :
( ( combine_options @ A @ F3 @ X4 @ ( none @ A ) )
= X4 ) ).
% combine_options_simps(2)
thf(fact_150_combine__options__simps_I1_J,axiom,
! [A: $tType,F3: A > A > A,Y2: option @ A] :
( ( combine_options @ A @ F3 @ ( none @ A ) @ Y2 )
= Y2 ) ).
% combine_options_simps(1)
thf(fact_151_combine__options__simps_I3_J,axiom,
! [A: $tType,F3: A > A > A,A2: A,B2: A] :
( ( combine_options @ A @ F3 @ ( some @ A @ A2 ) @ ( some @ A @ B2 ) )
= ( some @ A @ ( F3 @ A2 @ B2 ) ) ) ).
% combine_options_simps(3)
thf(fact_152_Differentialo_Osimps_I1_J,axiom,
! [Theta2: trm] :
( ( uSubst259074819ntialo @ ( some @ trm @ Theta2 ) )
= ( some @ trm @ ( differential @ Theta2 ) ) ) ).
% Differentialo.simps(1)
thf(fact_153_trm_Odistinct_I21_J,axiom,
! [X22: real,X7: trm] :
( ( number @ X22 )
!= ( differential @ X7 ) ) ).
% trm.distinct(21)
thf(fact_154_trm_Odistinct_I11_J,axiom,
! [X12: variable,X7: trm] :
( ( var @ X12 )
!= ( differential @ X7 ) ) ).
% trm.distinct(11)
thf(fact_155_trm_Odistinct_I1_J,axiom,
! [X12: variable,X22: real] :
( ( var @ X12 )
!= ( number @ X22 ) ) ).
% trm.distinct(1)
thf(fact_156_combine__options__assoc,axiom,
! [A: $tType,F3: A > A > A,X4: option @ A,Y2: option @ A,Z2: option @ A] :
( ! [X2: A,Y4: A,Z4: A] :
( ( F3 @ ( F3 @ X2 @ Y4 ) @ Z4 )
= ( F3 @ X2 @ ( F3 @ Y4 @ Z4 ) ) )
=> ( ( combine_options @ A @ F3 @ ( combine_options @ A @ F3 @ X4 @ Y2 ) @ Z2 )
= ( combine_options @ A @ F3 @ X4 @ ( combine_options @ A @ F3 @ Y2 @ Z2 ) ) ) ) ).
% combine_options_assoc
thf(fact_157_combine__options__commute,axiom,
! [A: $tType,F3: A > A > A,X4: option @ A,Y2: option @ A] :
( ! [X2: A,Y4: A] :
( ( F3 @ X2 @ Y4 )
= ( F3 @ Y4 @ X2 ) )
=> ( ( combine_options @ A @ F3 @ X4 @ Y2 )
= ( combine_options @ A @ F3 @ Y2 @ X4 ) ) ) ).
% combine_options_commute
thf(fact_158_combine__options__left__commute,axiom,
! [A: $tType,F3: A > A > A,Y2: option @ A,X4: option @ A,Z2: option @ A] :
( ! [X2: A,Y4: A] :
( ( F3 @ X2 @ Y4 )
= ( F3 @ Y4 @ X2 ) )
=> ( ! [X2: A,Y4: A,Z4: A] :
( ( F3 @ ( F3 @ X2 @ Y4 ) @ Z4 )
= ( F3 @ X2 @ ( F3 @ Y4 @ Z4 ) ) )
=> ( ( combine_options @ A @ F3 @ Y2 @ ( combine_options @ A @ F3 @ X4 @ Z2 ) )
= ( combine_options @ A @ F3 @ X4 @ ( combine_options @ A @ F3 @ Y2 @ Z2 ) ) ) ) ) ).
% combine_options_left_commute
thf(fact_159_usubstappt_Osimps_I7_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,Theta2: trm] :
( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( differential @ Theta2 ) )
= ( uSubst259074819ntialo
@ ( uSubst95898992stappt @ Sigma2
@ ( collect @ variable
@ ^ [X: variable] : $true )
@ Theta2 ) ) ) ).
% usubstappt.simps(7)
thf(fact_160_trm_Odistinct_I13_J,axiom,
! [X22: real,X3: char] :
( ( number @ X22 )
!= ( const @ X3 ) ) ).
% trm.distinct(13)
thf(fact_161_trm_Odistinct_I15_J,axiom,
! [X22: real,X41: char,X42: trm] :
( ( number @ X22 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(15)
thf(fact_162_trm_Odistinct_I19_J,axiom,
! [X22: real,X61: trm,X62: trm] :
( ( number @ X22 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(19)
thf(fact_163_trm_Odistinct_I17_J,axiom,
! [X22: real,X51: trm,X52: trm] :
( ( number @ X22 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(17)
thf(fact_164_Differentialo_Oelims,axiom,
! [X4: option @ trm,Y2: option @ trm] :
( ( ( uSubst259074819ntialo @ X4 )
= Y2 )
=> ( ! [Theta: trm] :
( ( X4
= ( some @ trm @ Theta ) )
=> ( Y2
!= ( some @ trm @ ( differential @ Theta ) ) ) )
=> ~ ( ( X4
= ( none @ trm ) )
=> ( Y2
!= ( none @ trm ) ) ) ) ) ).
% Differentialo.elims
thf(fact_165_trm_Odistinct_I3_J,axiom,
! [X12: variable,X3: char] :
( ( var @ X12 )
!= ( const @ X3 ) ) ).
% trm.distinct(3)
thf(fact_166_trm_Odistinct_I5_J,axiom,
! [X12: variable,X41: char,X42: trm] :
( ( var @ X12 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(5)
thf(fact_167_trm_Odistinct_I9_J,axiom,
! [X12: variable,X61: trm,X62: trm] :
( ( var @ X12 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(9)
thf(fact_168_trm_Odistinct_I7_J,axiom,
! [X12: variable,X51: trm,X52: trm] :
( ( var @ X12 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(7)
thf(fact_169_trm_Odistinct_I29_J,axiom,
! [X3: char,X7: trm] :
( ( const @ X3 )
!= ( differential @ X7 ) ) ).
% trm.distinct(29)
thf(fact_170_trm_Odistinct_I35_J,axiom,
! [X41: char,X42: trm,X7: trm] :
( ( func @ X41 @ X42 )
!= ( differential @ X7 ) ) ).
% trm.distinct(35)
thf(fact_171_trm_Odistinct_I41_J,axiom,
! [X61: trm,X62: trm,X7: trm] :
( ( times @ X61 @ X62 )
!= ( differential @ X7 ) ) ).
% trm.distinct(41)
thf(fact_172_trm_Odistinct_I39_J,axiom,
! [X51: trm,X52: trm,X7: trm] :
( ( plus @ X51 @ X52 )
!= ( differential @ X7 ) ) ).
% trm.distinct(39)
thf(fact_173_trm_Oexhaust,axiom,
! [Y2: trm] :
( ! [X13: variable] :
( Y2
!= ( var @ X13 ) )
=> ( ! [X23: real] :
( Y2
!= ( number @ X23 ) )
=> ( ! [X32: char] :
( Y2
!= ( const @ X32 ) )
=> ( ! [X412: char,X422: trm] :
( Y2
!= ( func @ X412 @ X422 ) )
=> ( ! [X512: trm,X522: trm] :
( Y2
!= ( plus @ X512 @ X522 ) )
=> ( ! [X612: trm,X622: trm] :
( Y2
!= ( times @ X612 @ X622 ) )
=> ~ ! [X72: trm] :
( Y2
!= ( differential @ X72 ) ) ) ) ) ) ) ) ).
% trm.exhaust
thf(fact_174_trm_Oinduct,axiom,
! [P: trm > $o,Trm: trm] :
( ! [X2: variable] : ( P @ ( var @ X2 ) )
=> ( ! [X2: real] : ( P @ ( number @ X2 ) )
=> ( ! [X2: char] : ( P @ ( const @ X2 ) )
=> ( ! [X1a: char,X2a2: trm] :
( ( P @ X2a2 )
=> ( P @ ( func @ X1a @ X2a2 ) ) )
=> ( ! [X1a: trm,X2a2: trm] :
( ( P @ X1a )
=> ( ( P @ X2a2 )
=> ( P @ ( plus @ X1a @ X2a2 ) ) ) )
=> ( ! [X1a: trm,X2a2: trm] :
( ( P @ X1a )
=> ( ( P @ X2a2 )
=> ( P @ ( times @ X1a @ X2a2 ) ) ) )
=> ( ! [X2: trm] :
( ( P @ X2 )
=> ( P @ ( differential @ X2 ) ) )
=> ( P @ Trm ) ) ) ) ) ) ) ) ).
% trm.induct
thf(fact_175_Geqo_Osimps_I2_J,axiom,
! [Eta2: option @ trm] :
( ( uSubst1556497037e_Geqo @ ( none @ trm ) @ Eta2 )
= ( none @ fml ) ) ).
% Geqo.simps(2)
thf(fact_176_Geqo__undef,axiom,
! [Theta2: option @ trm,Eta2: option @ trm] :
( ( ( uSubst1556497037e_Geqo @ Theta2 @ Eta2 )
= ( none @ fml ) )
= ( ( Theta2
= ( none @ trm ) )
| ( Eta2
= ( none @ trm ) ) ) ) ).
% Geqo_undef
thf(fact_177_Differentialo_Osimps_I2_J,axiom,
( ( uSubst259074819ntialo @ ( none @ trm ) )
= ( none @ trm ) ) ).
% Differentialo.simps(2)
thf(fact_178_Differentialo__undef,axiom,
! [Theta2: option @ trm] :
( ( ( uSubst259074819ntialo @ Theta2 )
= ( none @ trm ) )
= ( Theta2
= ( none @ trm ) ) ) ).
% Differentialo_undef
thf(fact_179_usubstappt_Osimps_I2_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,R2: real] :
( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( number @ R2 ) )
= ( some @ trm @ ( number @ R2 ) ) ) ).
% usubstappt.simps(2)
thf(fact_180_usubstappt_Osimps_I1_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,X4: variable] :
( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( var @ X4 ) )
= ( some @ trm @ ( var @ X4 ) ) ) ).
% usubstappt.simps(1)
thf(fact_181_usubstappt__differential__conv,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,Theta2: trm] :
( ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( differential @ Theta2 ) )
!= ( none @ trm ) )
=> ( ( uSubst95898992stappt @ Sigma2
@ ( collect @ variable
@ ^ [X: variable] : $true )
@ Theta2 )
!= ( none @ trm ) ) ) ).
% usubstappt_differential_conv
thf(fact_182_usubstappt_Opelims,axiom,
! [X4: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),Xa: set @ variable,Xb: trm,Y2: option @ trm] :
( ( ( uSubst95898992stappt @ X4 @ Xa @ Xb )
= Y2 )
=> ( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ X4 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ Xb ) ) )
=> ( ! [X2: variable] :
( ( Xb
= ( var @ X2 ) )
=> ( ( Y2
= ( some @ trm @ ( var @ X2 ) ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ X4 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ ( var @ X2 ) ) ) ) ) )
=> ( ! [R3: real] :
( ( Xb
= ( number @ R3 ) )
=> ( ( Y2
= ( some @ trm @ ( number @ R3 ) ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ X4 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ ( number @ R3 ) ) ) ) ) )
=> ( ! [F5: char] :
( ( Xb
= ( const @ F5 ) )
=> ( ( Y2
= ( uSubst1138577137pconst @ X4 @ Xa @ F5 ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ X4 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ ( const @ F5 ) ) ) ) ) )
=> ( ! [F5: char,Theta: trm] :
( ( Xb
= ( func @ F5 @ Theta ) )
=> ( ( Y2
= ( case_option @ ( option @ trm ) @ trm @ ( none @ trm )
@ ^ [Sigma_theta2: trm] :
( case_option @ ( option @ trm ) @ trm @ ( some @ trm @ ( func @ F5 @ Sigma_theta2 ) )
@ ^ [R: trm] :
( if @ ( option @ trm )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R ) @ Xa )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898992stappt @ ( uSubst969145931substt @ Sigma_theta2 ) @ ( 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 ) )
@ X4
@ F5 ) )
@ ( uSubst95898992stappt @ X4 @ Xa @ Theta ) ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ X4 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ ( func @ F5 @ Theta ) ) ) ) ) )
=> ( ! [Theta: trm,Eta: trm] :
( ( Xb
= ( plus @ Theta @ Eta ) )
=> ( ( Y2
= ( uSubst1112714340_Pluso @ ( uSubst95898992stappt @ X4 @ Xa @ Theta ) @ ( uSubst95898992stappt @ X4 @ Xa @ Eta ) ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ X4 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ ( plus @ Theta @ Eta ) ) ) ) ) )
=> ( ! [Theta: trm,Eta: trm] :
( ( Xb
= ( times @ Theta @ Eta ) )
=> ( ( Y2
= ( uSubst277968634Timeso @ ( uSubst95898992stappt @ X4 @ Xa @ Theta ) @ ( uSubst95898992stappt @ X4 @ Xa @ Eta ) ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ X4 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ ( times @ Theta @ Eta ) ) ) ) ) )
=> ~ ! [Theta: trm] :
( ( Xb
= ( differential @ Theta ) )
=> ( ( Y2
= ( uSubst259074819ntialo
@ ( uSubst95898992stappt @ X4
@ ( collect @ variable
@ ^ [X: variable] : $true )
@ Theta ) ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ X4 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ ( differential @ Theta ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% usubstappt.pelims
thf(fact_183_term__sem_Ocases,axiom,
! [X4: product_prod @ denotational_interp @ trm] :
( ! [I: denotational_interp,X2: variable] :
( X4
!= ( product_Pair @ denotational_interp @ trm @ I @ ( var @ X2 ) ) )
=> ( ! [I: denotational_interp,R3: real] :
( X4
!= ( product_Pair @ denotational_interp @ trm @ I @ ( number @ R3 ) ) )
=> ( ! [I: denotational_interp,F5: char] :
( X4
!= ( product_Pair @ denotational_interp @ trm @ I @ ( const @ F5 ) ) )
=> ( ! [I: denotational_interp,F5: char,Theta: trm] :
( X4
!= ( product_Pair @ denotational_interp @ trm @ I @ ( func @ F5 @ Theta ) ) )
=> ( ! [I: denotational_interp,Theta: trm,Eta: trm] :
( X4
!= ( product_Pair @ denotational_interp @ trm @ I @ ( plus @ Theta @ Eta ) ) )
=> ( ! [I: denotational_interp,Theta: trm,Eta: trm] :
( X4
!= ( product_Pair @ denotational_interp @ trm @ I @ ( times @ Theta @ Eta ) ) )
=> ~ ! [I: denotational_interp,Theta: trm] :
( X4
!= ( product_Pair @ denotational_interp @ trm @ I @ ( differential @ Theta ) ) ) ) ) ) ) ) ) ).
% term_sem.cases
thf(fact_184_term__sem_Oinduct,axiom,
! [P: denotational_interp > trm > $o,A0: denotational_interp,A1: trm] :
( ! [I: denotational_interp,X2: variable] : ( P @ I @ ( var @ X2 ) )
=> ( ! [I: denotational_interp,R3: real] : ( P @ I @ ( number @ R3 ) )
=> ( ! [I: denotational_interp,F5: char] : ( P @ I @ ( const @ F5 ) )
=> ( ! [I: denotational_interp,F5: char,Theta: trm] :
( ( P @ I @ Theta )
=> ( P @ I @ ( func @ F5 @ Theta ) ) )
=> ( ! [I: denotational_interp,Theta: trm] :
( ( P @ I @ Theta )
=> ! [Eta: trm] :
( ( P @ I @ Eta )
=> ( P @ I @ ( plus @ Theta @ Eta ) ) ) )
=> ( ! [I: denotational_interp,Theta: trm] :
( ( P @ I @ Theta )
=> ! [Eta: trm] :
( ( P @ I @ Eta )
=> ( P @ I @ ( times @ Theta @ Eta ) ) ) )
=> ( ! [I: denotational_interp,Theta: trm] :
( ( ? [Xa2: char] :
( member @ char @ Xa2
@ ( collect @ char
@ ^ [Uu: char] : $true ) )
=> ( P @ I @ Theta ) )
=> ( P @ I @ ( differential @ Theta ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ).
% term_sem.induct
thf(fact_185_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B7 ) )
= ( ( A2 = A7 )
& ( B2 = B7 ) ) ) ).
% old.prod.inject
thf(fact_186_prod_Oinject,axiom,
! [A: $tType,B: $tType,X12: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X12 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X12 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_187_case__prodI,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A2: A,B2: B] :
( ( F3 @ A2 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_188_case__prodI2,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,C2: A > B > $o] :
( ! [A5: A,B5: B] :
( ( P4
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( C2 @ A5 @ B5 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P4 ) ) ).
% case_prodI2
thf(fact_189_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P4: product_prod @ A @ B,C2: A > B > C > $o,X4: C] :
( ! [A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A5 @ B5 )
= P4 )
=> ( C2 @ A5 @ B5 @ X4 ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ X4 ) ) ).
% case_prodI2'
thf(fact_190_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
( ( member @ A @ Z2 @ ( C2 @ A2 @ B2 ) )
=> ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_191_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P4: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
( ! [A5: A,B5: B] :
( ( P4
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( member @ C @ Z2 @ ( C2 @ A5 @ B5 ) ) )
=> ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_192_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > C > A,A2: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F3 @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_193_usubstappt_Opsimps_I7_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,Theta2: trm] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma2 @ ( product_Pair @ ( set @ variable ) @ trm @ U2 @ ( differential @ Theta2 ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( differential @ Theta2 ) )
= ( uSubst259074819ntialo
@ ( uSubst95898992stappt @ Sigma2
@ ( collect @ variable
@ ^ [X: variable] : $true )
@ Theta2 ) ) ) ) ).
% usubstappt.psimps(7)
thf(fact_194_usubstappt_Opsimps_I1_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,X4: variable] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma2 @ ( product_Pair @ ( set @ variable ) @ trm @ U2 @ ( var @ X4 ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( var @ X4 ) )
= ( some @ trm @ ( var @ X4 ) ) ) ) ).
% usubstappt.psimps(1)
thf(fact_195_usubstappt_Opsimps_I2_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,R2: real] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma2 @ ( product_Pair @ ( set @ variable ) @ trm @ U2 @ ( number @ R2 ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( number @ R2 ) )
= ( some @ trm @ ( number @ R2 ) ) ) ) ).
% usubstappt.psimps(2)
thf(fact_196_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C,G2: ( product_prod @ A @ B ) > C] :
( ! [X2: A,Y4: B] :
( ( F3 @ X2 @ Y4 )
= ( G2 @ ( product_Pair @ A @ B @ X2 @ Y4 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F3 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_197_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X: A,Y: B] : ( F3 @ ( product_Pair @ A @ B @ X @ Y ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_198_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z2: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) )
=> ~ ! [X2: B,Y4: C] :
( ( Z2
= ( product_Pair @ B @ C @ X2 @ Y4 ) )
=> ~ ( Q @ ( P @ X2 @ Y4 ) ) ) ) ).
% case_prodE2
thf(fact_199_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R4 )
@ ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ S ) )
= ( ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S ) ) ) ) ).
% inf_Int_eq2
thf(fact_200_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_201_case__prodD,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A2: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( F3 @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_202_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P4: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P4 )
=> ~ ! [X2: A,Y4: B] :
( ( P4
= ( product_Pair @ A @ B @ X2 @ Y4 ) )
=> ~ ( C2 @ X2 @ Y4 ) ) ) ).
% case_prodE
thf(fact_203_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R4: A > B > C > $o,A2: A,B2: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R4 @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
=> ( R4 @ A2 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_204_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P4: product_prod @ A @ B,Z2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ Z2 )
=> ~ ! [X2: A,Y4: B] :
( ( P4
= ( product_Pair @ A @ B @ X2 @ Y4 ) )
=> ~ ( C2 @ X2 @ Y4 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_205_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R4 ) )
= ( ^ [X: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ S ) ) )
= ( R4 = S ) ) ).
% pred_equals_eq2
thf(fact_206_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P4: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P4 ) )
=> ~ ! [X2: B,Y4: C] :
( ( P4
= ( product_Pair @ B @ C @ X2 @ Y4 ) )
=> ~ ( member @ A @ Z2 @ ( C2 @ X2 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_207_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_208_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y2: product_prod @ A @ B] :
~ ! [A5: A,B5: B] :
( Y2
!= ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_209_prod__induct7,axiom,
! [G3: $tType,F6: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) )] :
( ! [A5: A,B5: B,C4: C,D2: D,E2: E,F5: F6,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F6 @ G3 ) @ E2 @ ( product_Pair @ F6 @ G3 @ F5 @ G4 ) ) ) ) ) ) )
=> ( P @ X4 ) ) ).
% prod_induct7
thf(fact_210_prod__induct6,axiom,
! [F6: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) ) )] :
( ! [A5: A,B5: B,C4: C,D2: D,E2: E,F5: F6] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F6 ) @ D2 @ ( product_Pair @ E @ F6 @ E2 @ F5 ) ) ) ) ) )
=> ( P @ X4 ) ) ).
% prod_induct6
thf(fact_211_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B5: B,C4: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X4 ) ) ).
% prod_induct5
thf(fact_212_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B5: B,C4: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) )
=> ( P @ X4 ) ) ).
% prod_induct4
thf(fact_213_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B5: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C4 ) ) )
=> ( P @ X4 ) ) ).
% prod_induct3
thf(fact_214_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F6: $tType,G3: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) )] :
~ ! [A5: A,B5: B,C4: C,D2: D,E2: E,F5: F6,G4: G3] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F6 @ G3 ) @ E2 @ ( product_Pair @ F6 @ G3 @ F5 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_215_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F6: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) ) )] :
~ ! [A5: A,B5: B,C4: C,D2: D,E2: E,F5: F6] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F6 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F6 ) @ D2 @ ( product_Pair @ E @ F6 @ E2 @ F5 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_216_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A5: A,B5: B,C4: C,D2: D,E2: E] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_217_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B5: B,C4: C,D2: D] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_218_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y2: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B5: B,C4: C] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C4 ) ) ) ).
% prod_cases3
thf(fact_219_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A7: A,B7: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B7 ) )
=> ~ ( ( A2 = A7 )
=> ( B2 != B7 ) ) ) ).
% Pair_inject
thf(fact_220_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_221_surj__pair,axiom,
! [A: $tType,B: $tType,P4: product_prod @ A @ B] :
? [X2: A,Y4: B] :
( P4
= ( product_Pair @ A @ B @ X2 @ Y4 ) ) ).
% surj_pair
thf(fact_222_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > B > C,X12: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F3 @ ( product_Pair @ A @ B @ X12 @ X22 ) )
= ( F3 @ X12 @ X22 ) ) ).
% old.prod.case
thf(fact_223_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P4 )
= P4 ) ).
% case_prod_Pair_iden
thf(fact_224_usubstappt_Opsimps_I3_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,F3: char] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma2 @ ( product_Pair @ ( set @ variable ) @ trm @ U2 @ ( const @ F3 ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( const @ F3 ) )
= ( uSubst1138577137pconst @ Sigma2 @ U2 @ F3 ) ) ) ).
% usubstappt.psimps(3)
thf(fact_225_usubstappt_Opsimps_I6_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,Theta2: trm,Eta2: trm] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma2 @ ( product_Pair @ ( set @ variable ) @ trm @ U2 @ ( times @ Theta2 @ Eta2 ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( times @ Theta2 @ Eta2 ) )
= ( uSubst277968634Timeso @ ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 ) @ ( uSubst95898992stappt @ Sigma2 @ U2 @ Eta2 ) ) ) ) ).
% usubstappt.psimps(6)
thf(fact_226_usubstappt_Opsimps_I5_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,Theta2: trm,Eta2: trm] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma2 @ ( product_Pair @ ( set @ variable ) @ trm @ U2 @ ( plus @ Theta2 @ Eta2 ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( plus @ Theta2 @ Eta2 ) )
= ( uSubst1112714340_Pluso @ ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 ) @ ( uSubst95898992stappt @ Sigma2 @ U2 @ Eta2 ) ) ) ) ).
% usubstappt.psimps(5)
thf(fact_227_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B2: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( C2 @ A2 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_228_usubstappt_Opsimps_I4_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,F3: char,Theta2: trm] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma2 @ ( product_Pair @ ( set @ variable ) @ trm @ U2 @ ( func @ F3 @ Theta2 ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma2 @ U2 @ ( func @ F3 @ Theta2 ) )
= ( case_option @ ( option @ trm ) @ trm @ ( none @ trm )
@ ^ [Sigma_theta2: trm] :
( case_option @ ( option @ trm ) @ trm @ ( some @ trm @ ( func @ F3 @ Sigma_theta2 ) )
@ ^ [R: trm] :
( if @ ( option @ trm )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R ) @ U2 )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898992stappt @ ( uSubst969145931substt @ Sigma_theta2 ) @ ( 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 ) )
@ Sigma2
@ F3 ) )
@ ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 ) ) ) ) ).
% usubstappt.psimps(4)
thf(fact_229_usubstappt_Ocases,axiom,
! [X4: product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm )] :
( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,X2: variable] :
( X4
!= ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( var @ X2 ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,R3: real] :
( X4
!= ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( number @ R3 ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,F5: char] :
( X4
!= ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( const @ F5 ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,F5: char,Theta: trm] :
( X4
!= ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( func @ F5 @ Theta ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta: trm,Eta: trm] :
( X4
!= ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( plus @ Theta @ Eta ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta: trm,Eta: trm] :
( X4
!= ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( times @ Theta @ Eta ) ) ) )
=> ~ ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta: trm] :
( X4
!= ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( differential @ Theta ) ) ) ) ) ) ) ) ) ) ).
% usubstappt.cases
thf(fact_230_usubstappt_Opinduct,axiom,
! [A0: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),A1: set @ variable,A22: trm,P: ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) > ( set @ variable ) > trm > $o] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ A0 @ ( product_Pair @ ( set @ variable ) @ trm @ A1 @ A22 ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,X2: variable] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( var @ X2 ) ) ) )
=> ( P @ Sigma3 @ U3 @ ( var @ X2 ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,R3: real] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( number @ R3 ) ) ) )
=> ( P @ Sigma3 @ U3 @ ( number @ R3 ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,F5: char] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( const @ F5 ) ) ) )
=> ( P @ Sigma3 @ U3 @ ( const @ F5 ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,F5: char,Theta: trm] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( func @ F5 @ Theta ) ) ) )
=> ( ( P @ Sigma3 @ U3 @ Theta )
=> ( ! [X25: trm] :
( ( ( uSubst95898992stappt @ Sigma3 @ U3 @ Theta )
= ( some @ trm @ X25 ) )
=> ! [X2a: 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 ) )
@ Sigma3
@ F5 )
= ( some @ trm @ X2a ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ X2a ) @ U3 )
= ( bot_bot @ ( set @ variable ) ) )
=> ( P @ ( uSubst969145931substt @ X25 ) @ ( bot_bot @ ( set @ variable ) ) @ X2a ) ) ) )
=> ( P @ Sigma3 @ U3 @ ( func @ F5 @ Theta ) ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta: trm,Eta: trm] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( plus @ Theta @ Eta ) ) ) )
=> ( ( P @ Sigma3 @ U3 @ Theta )
=> ( ( P @ Sigma3 @ U3 @ Eta )
=> ( P @ Sigma3 @ U3 @ ( plus @ Theta @ Eta ) ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta: trm,Eta: trm] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( times @ Theta @ Eta ) ) ) )
=> ( ( P @ Sigma3 @ U3 @ Theta )
=> ( ( P @ Sigma3 @ U3 @ Eta )
=> ( P @ Sigma3 @ U3 @ ( times @ Theta @ Eta ) ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta: trm] :
( ( accp @ ( product_prod @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) ) @ uSubst2096773001pt_rel @ ( product_Pair @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ) @ ( product_prod @ ( set @ variable ) @ trm ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( differential @ Theta ) ) ) )
=> ( ( P @ Sigma3
@ ( collect @ variable
@ ^ [X: variable] : $true )
@ Theta )
=> ( P @ Sigma3 @ U3 @ ( differential @ Theta ) ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ) ) ) ) ) ).
% usubstappt.pinduct
thf(fact_231_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_232_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q3: product_prod @ A @ B,F3: A > B > C,G2: A > B > C,P4: product_prod @ A @ B] :
( ! [X2: A,Y4: B] :
( ( ( product_Pair @ A @ B @ X2 @ Y4 )
= Q3 )
=> ( ( F3 @ X2 @ Y4 )
= ( G2 @ X2 @ Y4 ) ) )
=> ( ( P4 = Q3 )
=> ( ( product_case_prod @ A @ B @ C @ F3 @ P4 )
= ( product_case_prod @ A @ B @ C @ G2 @ Q3 ) ) ) ) ).
% split_cong
thf(fact_233_Geqo_Oelims,axiom,
! [X4: option @ trm,Xa: option @ trm,Y2: option @ fml] :
( ( ( uSubst1556497037e_Geqo @ X4 @ Xa )
= Y2 )
=> ( ! [Theta: trm] :
( ( X4
= ( some @ trm @ Theta ) )
=> ! [Eta: trm] :
( ( Xa
= ( some @ trm @ Eta ) )
=> ( Y2
!= ( some @ fml @ ( geq @ Theta @ Eta ) ) ) ) )
=> ( ( ( X4
= ( none @ trm ) )
=> ( Y2
!= ( none @ fml ) ) )
=> ~ ( ? [V: trm] :
( X4
= ( some @ trm @ V ) )
=> ( ( Xa
= ( none @ trm ) )
=> ( Y2
!= ( none @ fml ) ) ) ) ) ) ) ).
% Geqo.elims
thf(fact_234_fml_Oinject_I2_J,axiom,
! [X21: trm,X222: trm,Y21: trm,Y222: trm] :
( ( ( geq @ X21 @ X222 )
= ( geq @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% fml.inject(2)
thf(fact_235_Diamondo_Ocases,axiom,
! [X4: product_prod @ ( option @ game ) @ ( option @ fml )] :
( ! [Alpha: game,Phi: fml] :
( X4
!= ( product_Pair @ ( option @ game ) @ ( option @ fml ) @ ( some @ game @ Alpha ) @ ( some @ fml @ Phi ) ) )
=> ( ! [Phi: option @ fml] :
( X4
!= ( product_Pair @ ( option @ game ) @ ( option @ fml ) @ ( none @ game ) @ Phi ) )
=> ~ ! [V: game] :
( X4
!= ( product_Pair @ ( option @ game ) @ ( option @ fml ) @ ( some @ game @ V ) @ ( none @ fml ) ) ) ) ) ).
% Diamondo.cases
thf(fact_236_Existso_Ocases,axiom,
! [X4: product_prod @ variable @ ( option @ fml )] :
( ! [X2: variable,Phi: fml] :
( X4
!= ( product_Pair @ variable @ ( option @ fml ) @ X2 @ ( some @ fml @ Phi ) ) )
=> ~ ! [X2: variable] :
( X4
!= ( product_Pair @ variable @ ( option @ fml ) @ X2 @ ( none @ fml ) ) ) ) ).
% Existso.cases
thf(fact_237_Ando_Ocases,axiom,
! [X4: product_prod @ ( option @ fml ) @ ( option @ fml )] :
( ! [Phi: fml,Psi: fml] :
( X4
!= ( product_Pair @ ( option @ fml ) @ ( option @ fml ) @ ( some @ fml @ Phi ) @ ( some @ fml @ Psi ) ) )
=> ( ! [Psi: option @ fml] :
( X4
!= ( product_Pair @ ( option @ fml ) @ ( option @ fml ) @ ( none @ fml ) @ Psi ) )
=> ~ ! [V: fml] :
( X4
!= ( product_Pair @ ( option @ fml ) @ ( option @ fml ) @ ( some @ fml @ V ) @ ( none @ fml ) ) ) ) ) ).
% Ando.cases
thf(fact_238_ODEo_Ocases,axiom,
! [X4: product_prod @ char @ ( option @ trm )] :
( ! [X2: char,Theta: trm] :
( X4
!= ( product_Pair @ char @ ( option @ trm ) @ X2 @ ( some @ trm @ Theta ) ) )
=> ~ ! [X2: char] :
( X4
!= ( product_Pair @ char @ ( option @ trm ) @ X2 @ ( none @ trm ) ) ) ) ).
% ODEo.cases
thf(fact_239_Assigno_Ocases,axiom,
! [X4: product_prod @ variable @ ( option @ trm )] :
( ! [X2: variable,Theta: trm] :
( X4
!= ( product_Pair @ variable @ ( option @ trm ) @ X2 @ ( some @ trm @ Theta ) ) )
=> ~ ! [X2: variable] :
( X4
!= ( product_Pair @ variable @ ( option @ trm ) @ X2 @ ( none @ trm ) ) ) ) ).
% Assigno.cases
thf(fact_240_Timeso_Ocases,axiom,
! [X4: product_prod @ ( option @ trm ) @ ( option @ trm )] :
( ! [Theta: trm,Eta: trm] :
( X4
!= ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ Theta ) @ ( some @ trm @ Eta ) ) )
=> ( ! [Eta: option @ trm] :
( X4
!= ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( none @ trm ) @ Eta ) )
=> ~ ! [V: trm] :
( X4
!= ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ V ) @ ( none @ trm ) ) ) ) ) ).
% Timeso.cases
thf(fact_241_Geqo_Osimps_I1_J,axiom,
! [Theta2: trm,Eta2: trm] :
( ( uSubst1556497037e_Geqo @ ( some @ trm @ Theta2 ) @ ( some @ trm @ Eta2 ) )
= ( some @ fml @ ( geq @ Theta2 @ Eta2 ) ) ) ).
% Geqo.simps(1)
thf(fact_242_usubstappf__geqr,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,Theta2: trm,Eta2: trm] :
( ( ( uSubst95898978stappf @ Sigma2 @ U2 @ ( geq @ Theta2 @ Eta2 ) )
!= ( none @ fml ) )
=> ( ( uSubst95898978stappf @ Sigma2 @ U2 @ ( geq @ Theta2 @ Eta2 ) )
= ( some @ fml @ ( geq @ ( the @ trm @ ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 ) ) @ ( the @ trm @ ( uSubst95898992stappt @ Sigma2 @ U2 @ Eta2 ) ) ) ) ) ) ).
% usubstappf_geqr
thf(fact_243_usubstappf__geq,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,Theta2: trm,Eta2: trm] :
( ( ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma2 @ U2 @ Eta2 )
!= ( none @ trm ) )
=> ( ( uSubst95898978stappf @ Sigma2 @ U2 @ ( geq @ Theta2 @ Eta2 ) )
= ( some @ fml @ ( geq @ ( the @ trm @ ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 ) ) @ ( the @ trm @ ( uSubst95898992stappt @ Sigma2 @ U2 @ Eta2 ) ) ) ) ) ) ) ).
% usubstappf_geq
thf(fact_244_Composeo_Ocases,axiom,
! [X4: product_prod @ ( option @ game ) @ ( option @ game )] :
( ! [Alpha: game,Beta: game] :
( X4
!= ( product_Pair @ ( option @ game ) @ ( option @ game ) @ ( some @ game @ Alpha ) @ ( some @ game @ Beta ) ) )
=> ( ! [Alpha: option @ game] :
( X4
!= ( product_Pair @ ( option @ game ) @ ( option @ game ) @ Alpha @ ( none @ game ) ) )
=> ~ ! [V: game] :
( X4
!= ( product_Pair @ ( option @ game ) @ ( option @ game ) @ ( none @ game ) @ ( some @ game @ V ) ) ) ) ) ).
% Composeo.cases
thf(fact_245_usubstappf_Osimps_I2_J,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,Theta2: trm,Eta2: trm] :
( ( uSubst95898978stappf @ Sigma2 @ U2 @ ( geq @ Theta2 @ Eta2 ) )
= ( uSubst1556497037e_Geqo @ ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 ) @ ( uSubst95898992stappt @ Sigma2 @ U2 @ Eta2 ) ) ) ).
% usubstappf.simps(2)
thf(fact_246_usubstappf__geq__conv,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,Theta2: trm,Eta2: trm] :
( ( ( uSubst95898978stappf @ Sigma2 @ U2 @ ( geq @ Theta2 @ Eta2 ) )
!= ( none @ fml ) )
=> ( ( ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 )
!= ( none @ trm ) )
& ( ( uSubst95898992stappt @ Sigma2 @ U2 @ Eta2 )
!= ( none @ trm ) ) ) ) ).
% usubstappf_geq_conv
thf(fact_247_dotsubstt__def,axiom,
( uSubst969145931substt
= ( ^ [Theta3: trm] :
( product_Pair @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) )
@ ^ [F2: char] :
( if @ ( option @ trm )
@ ( F2
= ( char2 @ $false @ $true @ $true @ $true @ $false @ $true @ $false @ $false ) )
@ ( some @ trm @ Theta3 )
@ ( none @ trm ) )
@ ( product_Pair @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) )
@ ^ [Uu: char] : ( none @ trm )
@ ( product_Pair @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) )
@ ^ [Uu: char] : ( none @ fml )
@ ^ [Uu: char] : ( none @ game ) ) ) ) ) ) ).
% dotsubstt_def
thf(fact_248_Geqo_Opelims,axiom,
! [X4: option @ trm,Xa: option @ trm,Y2: option @ fml] :
( ( ( uSubst1556497037e_Geqo @ X4 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst864323244qo_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ X4 @ Xa ) )
=> ( ! [Theta: trm] :
( ( X4
= ( some @ trm @ Theta ) )
=> ! [Eta: trm] :
( ( Xa
= ( some @ trm @ Eta ) )
=> ( ( Y2
= ( some @ fml @ ( geq @ Theta @ Eta ) ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst864323244qo_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ Theta ) @ ( some @ trm @ Eta ) ) ) ) ) )
=> ( ( ( X4
= ( none @ trm ) )
=> ( ( Y2
= ( none @ fml ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst864323244qo_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( none @ trm ) @ Xa ) ) ) )
=> ~ ! [V: trm] :
( ( X4
= ( some @ trm @ V ) )
=> ( ( Xa
= ( none @ trm ) )
=> ( ( Y2
= ( none @ fml ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst864323244qo_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ V ) @ ( none @ trm ) ) ) ) ) ) ) ) ) ) ).
% Geqo.pelims
thf(fact_249_Timeso_Opelims,axiom,
! [X4: option @ trm,Xa: option @ trm,Y2: option @ trm] :
( ( ( uSubst277968634Timeso @ X4 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst1377811071so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ X4 @ Xa ) )
=> ( ! [Theta: trm] :
( ( X4
= ( some @ trm @ Theta ) )
=> ! [Eta: trm] :
( ( Xa
= ( some @ trm @ Eta ) )
=> ( ( Y2
= ( some @ trm @ ( times @ Theta @ Eta ) ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst1377811071so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ Theta ) @ ( some @ trm @ Eta ) ) ) ) ) )
=> ( ( ( X4
= ( none @ trm ) )
=> ( ( Y2
= ( none @ trm ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst1377811071so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( none @ trm ) @ Xa ) ) ) )
=> ~ ! [V: trm] :
( ( X4
= ( some @ trm @ V ) )
=> ( ( Xa
= ( none @ trm ) )
=> ( ( Y2
= ( none @ trm ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst1377811071so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ V ) @ ( none @ trm ) ) ) ) ) ) ) ) ) ) ).
% Timeso.pelims
thf(fact_250_Pluso_Opelims,axiom,
! [X4: option @ trm,Xa: option @ trm,Y2: option @ trm] :
( ( ( uSubst1112714340_Pluso @ X4 @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst270600597so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ X4 @ Xa ) )
=> ( ! [Theta: trm] :
( ( X4
= ( some @ trm @ Theta ) )
=> ! [Eta: trm] :
( ( Xa
= ( some @ trm @ Eta ) )
=> ( ( Y2
= ( some @ trm @ ( plus @ Theta @ Eta ) ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst270600597so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ Theta ) @ ( some @ trm @ Eta ) ) ) ) ) )
=> ( ( ( X4
= ( none @ trm ) )
=> ( ( Y2
= ( none @ trm ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst270600597so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( none @ trm ) @ Xa ) ) ) )
=> ~ ! [V: trm] :
( ( X4
= ( some @ trm @ V ) )
=> ( ( Xa
= ( none @ trm ) )
=> ( ( Y2
= ( none @ trm ) )
=> ~ ( accp @ ( product_prod @ ( option @ trm ) @ ( option @ trm ) ) @ uSubst270600597so_rel @ ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ V ) @ ( none @ trm ) ) ) ) ) ) ) ) ) ) ).
% Pluso.pelims
thf(fact_251_dot__def,axiom,
( uSubst_Mirabelle_dot
= ( const @ ( char2 @ $false @ $true @ $true @ $true @ $false @ $true @ $false @ $false ) ) ) ).
% dot_def
thf(fact_252_usubstappf__pred,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),P4: char,R2: fml,U2: set @ variable,Theta2: trm,Sigma_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 ) )
@ ^ [P3: char > ( option @ fml ),Uw: char > ( option @ game )] : P3 ) )
@ Sigma2
@ P4 )
= ( some @ fml @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ R2 ) @ U2 )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( ( uSubst95898992stappt @ Sigma2 @ U2 @ Theta2 )
= ( some @ trm @ Sigma_theta ) )
=> ( ( uSubst95898978stappf @ Sigma2 @ U2 @ ( pred @ P4 @ Theta2 ) )
= ( uSubst95898978stappf @ ( uSubst969145931substt @ Sigma_theta ) @ ( bot_bot @ ( set @ variable ) ) @ R2 ) ) ) ) ) ).
% usubstappf_pred
thf(fact_253_fml_Oinject_I1_J,axiom,
! [X11: char,X122: trm,Y11: char,Y12: trm] :
( ( ( pred @ X11 @ X122 )
= ( pred @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X122 = Y12 ) ) ) ).
% fml.inject(1)
thf(fact_254_usubstappf__pred2,axiom,
! [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),P4: char,R2: fml,U2: set @ variable,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 ) )
@ ^ [P3: char > ( option @ fml ),Uw: char > ( option @ game )] : P3 ) )
@ Sigma2
@ P4 )
= ( some @ fml @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ R2 ) @ U2 )
!= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst95898978stappf @ Sigma2 @ U2 @ ( pred @ P4 @ Theta2 ) )
= ( none @ fml ) ) ) ) ).
% usubstappf_pred2
% Type constructors (17)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounded_lattice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 )
=> ( bounded_lattice_bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_inf @ A9 )
=> ( semilattice_inf @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 )
=> ( lattice @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 )
=> ( bot @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_3,axiom,
! [A8: $tType] : ( bounded_lattice_bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_4,axiom,
! [A8: $tType] : ( semilattice_inf @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_5,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_6,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_7,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_8,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_9,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_10,axiom,
bot @ $o ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__inf_11,axiom,
semilattice_inf @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_12,axiom,
lattice @ real ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X4: A,Y2: A] :
( ( if @ A @ $false @ X4 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X4: A,Y2: A] :
( ( if @ A @ $true @ X4 @ Y2 )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ( 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
@ f )
= ( none @ trm ) )
| ( ( uSubst95898992stappt @ sigma @ u @ ( const @ f ) )
= ( uSubst1138577137pconst @ sigma @ v @ f ) ) ) ).
%------------------------------------------------------------------------------