TPTP Problem File: ITP203^2.p
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%------------------------------------------------------------------------------
% File : ITP203^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer USubst problem prob_534__6335794_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_534__6335794_1 [Des21]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 334 ( 154 unt; 58 typ; 0 def)
% Number of atoms : 608 ( 443 equ; 0 cnn)
% Maximal formula atoms : 31 ( 2 avg)
% Number of connectives : 7417 ( 175 ~; 10 |; 28 &;6919 @)
% ( 0 <=>; 285 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 1365 (1365 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 52 usr; 8 con; 0-8 aty)
% Number of variables : 1347 ( 269 ^;1035 !; 14 ?;1347 :)
% ( 29 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:19:55.955
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
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 (49)
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_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_OPluso,type,
uSubst1112714340_Pluso: ( option @ trm ) > ( option @ trm ) > ( option @ trm ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OTesto,type,
uSubst190403692_Testo: ( option @ fml ) > ( option @ game ) ).
thf(sy_c_USubst__Mirabelle__nnnzepxswx_OTimeso,type,
uSubst277968634Timeso: ( option @ trm ) > ( option @ trm ) > ( option @ trm ) ).
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_Ua____,type,
ua: set @ variable ).
thf(sy_v_Va____,type,
va: set @ variable ).
thf(sy_v__092_060sigma_062,type,
sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) ).
thf(sy_v__092_060theta_062____,type,
theta: trm ).
thf(sy_v_p____,type,
p: char ).
% Relevant facts (255)
thf(fact_0_f7,axiom,
! [Z: option @ fml,F: fml > ( option @ fml ),Za: option @ fml] :
( ( ( Za
= ( none @ fml ) )
=> ( ( case_option @ ( option @ fml ) @ fml @ Z @ F @ Za )
= Z ) )
& ( ( Za
!= ( none @ fml ) )
=> ( ( case_option @ ( option @ fml ) @ fml @ Z @ F @ Za )
= ( F @ ( the @ fml @ Za ) ) ) ) ) ).
% f7
thf(fact_1_Pred_Oprems_I2_J,axiom,
( ( uSubst95898978stappf @ sigma @ va @ ( pred @ p @ theta ) )
!= ( none @ fml ) ) ).
% Pred.prems(2)
thf(fact_2_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_3_Pred_Oprems_I1_J,axiom,
( ( uSubst95898978stappf @ sigma @ ua @ ( pred @ p @ theta ) )
!= ( none @ fml ) ) ).
% Pred.prems(1)
thf(fact_4__092_060open_062_Iif_AFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_A_092_060inter_062_AV_A_061_A_123_125_Athen_Ausubstappf_A_Idotsubstt_A_Ithe_A_Iusubstappt_A_092_060sigma_062_AV_A_092_060theta_062_J_J_J_A_123_125_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_Aelse_Aundeff_J_A_092_060noteq_062_Ausubstappf_A_Idotsubstt_A_Ithe_A_Iusubstappt_A_092_060sigma_062_AV_A_092_060theta_062_J_J_J_A_123_125_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_092_060close_062,axiom,
~ ( ( ( inf_inf @ ( set @ variable )
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) )
@ va )
!= ( bot_bot @ ( set @ variable ) ) )
=> ( ( none @ fml )
= ( uSubst95898978stappf @ ( uSubst969145931substt @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) @ ( bot_bot @ ( set @ variable ) )
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) ) ) ).
% \<open>(if FVF (the (SPreds \<sigma> p)) \<inter> V = {} then usubstappf (dotsubstt (the (usubstappt \<sigma> V \<theta>))) {} (the (SPreds \<sigma> p)) else undeff) \<noteq> usubstappf (dotsubstt (the (usubstappt \<sigma> V \<theta>))) {} (the (SPreds \<sigma> p))\<close>
thf(fact_5_f9,axiom,
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p )
= ( none @ fml ) )
=> ( ( uSubst95898978stappf @ sigma @ ua @ ( pred @ p @ theta ) )
= ( uSubst95898978stappf @ sigma @ va @ ( pred @ p @ theta ) ) ) ) ).
% f9
thf(fact_6__092_060open_062_Icase_ASPreds_A_092_060sigma_062_Ap_Aof_ANone_A_092_060Rightarrow_062_AAfml_A_IPred_Ap_A_Ithe_A_Iusubstappt_A_092_060sigma_062_AV_A_092_060theta_062_J_J_J_A_124_ASome_Af_A_092_060Rightarrow_062_Aif_AFVF_Af_A_092_060inter_062_AV_A_061_A_123_125_Athen_Ausubstappf_A_Idotsubstt_A_Ithe_A_Iusubstappt_A_092_060sigma_062_AV_A_092_060theta_062_J_J_J_A_123_125_Af_Aelse_Aundeff_J_A_092_060noteq_062_A_Iif_AFVF_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_A_092_060inter_062_AV_A_061_A_123_125_Athen_Ausubstappf_A_Idotsubstt_A_Ithe_A_Iusubstappt_A_092_060sigma_062_AV_A_092_060theta_062_J_J_J_A_123_125_A_Ithe_A_ISPreds_A_092_060sigma_062_Ap_J_J_Aelse_Aundeff_J_092_060close_062,axiom,
~ ( ( ( ( inf_inf @ ( set @ variable )
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) )
@ va )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ p @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) )
@ ^ [F2: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ F2 ) @ va )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) @ ( bot_bot @ ( set @ variable ) ) @ F2 )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) )
= ( uSubst95898978stappf @ ( uSubst969145931substt @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) @ ( bot_bot @ ( set @ variable ) )
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) ) )
& ( ( ( inf_inf @ ( set @ variable )
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) )
@ va )
!= ( bot_bot @ ( set @ variable ) ) )
=> ( ( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ p @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) )
@ ^ [F2: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ F2 ) @ va )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) @ ( bot_bot @ ( set @ variable ) ) @ F2 )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) )
= ( none @ fml ) ) ) ) ).
% \<open>(case SPreds \<sigma> p of None \<Rightarrow> Afml (Pred p (the (usubstappt \<sigma> V \<theta>))) | Some f \<Rightarrow> if FVF f \<inter> V = {} then usubstappf (dotsubstt (the (usubstappt \<sigma> V \<theta>))) {} f else undeff) \<noteq> (if FVF (the (SPreds \<sigma> p)) \<inter> V = {} then usubstappf (dotsubstt (the (usubstappt \<sigma> V \<theta>))) {} (the (SPreds \<sigma> p)) else undeff)\<close>
thf(fact_7_Ando_Oinduct,axiom,
! [P2: ( option @ fml ) > ( option @ fml ) > $o,A0: option @ fml,A1: option @ fml] :
( ! [Phi: fml,Psi: fml] : ( P2 @ ( some @ fml @ Phi ) @ ( some @ fml @ Psi ) )
=> ( ! [X_1: option @ fml] : ( P2 @ ( none @ fml ) @ X_1 )
=> ( ! [V: fml] : ( P2 @ ( some @ fml @ V ) @ ( none @ fml ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% Ando.induct
thf(fact_8_Testo_Ocases,axiom,
! [X2: option @ fml] :
( ! [Phi: fml] :
( X2
!= ( some @ fml @ Phi ) )
=> ( X2
= ( none @ fml ) ) ) ).
% Testo.cases
thf(fact_9_Testo_Oinduct,axiom,
! [P2: ( option @ fml ) > $o,A0: option @ fml] :
( ! [Phi: fml] : ( P2 @ ( some @ fml @ Phi ) )
=> ( ( P2 @ ( none @ fml ) )
=> ( P2 @ A0 ) ) ) ).
% Testo.induct
thf(fact_10_undeff__equiv,axiom,
! [Phi2: option @ fml] :
( ( Phi2
!= ( none @ fml ) )
= ( ? [F2: fml] :
( Phi2
= ( some @ fml @ F2 ) ) ) ) ).
% undeff_equiv
thf(fact_11_Existso_Oinduct,axiom,
! [P2: variable > ( option @ fml ) > $o,A0: variable,A1: option @ fml] :
( ! [X3: variable,Phi: fml] : ( P2 @ X3 @ ( some @ fml @ Phi ) )
=> ( ! [X3: variable] : ( P2 @ X3 @ ( none @ fml ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% Existso.induct
thf(fact_12_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,X22: B] : ( H @ ( F3 @ X1 @ X22 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_13__092_060open_062usubstappf_A_092_060sigma_062_AU_A_IPred_Ap_A_092_060theta_062_J_A_092_060noteq_062_Ausubstappf_A_092_060sigma_062_AV_A_IPred_Ap_A_092_060theta_062_J_092_060close_062,axiom,
( ( uSubst95898978stappf @ sigma @ ua @ ( pred @ p @ theta ) )
!= ( uSubst95898978stappf @ sigma @ va @ ( pred @ p @ theta ) ) ) ).
% \<open>usubstappf \<sigma> U (Pred p \<theta>) \<noteq> usubstappf \<sigma> V (Pred p \<theta>)\<close>
thf(fact_14_f5,axiom,
( ( uSubst95898992stappt @ sigma @ ua @ theta )
= ( uSubst95898992stappt @ sigma @ va @ theta ) ) ).
% f5
thf(fact_15_calculation_I1_J,axiom,
~ ( ( ( ( inf_inf @ ( set @ variable )
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) )
@ va )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst95898978stappf @ sigma @ ua @ ( pred @ p @ theta ) )
= ( uSubst95898978stappf @ ( uSubst969145931substt @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) @ ( bot_bot @ ( set @ variable ) )
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) ) )
& ( ( ( inf_inf @ ( set @ variable )
@ ( static_FVF
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) )
@ va )
!= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst95898978stappf @ sigma @ ua @ ( pred @ p @ theta ) )
= ( none @ fml ) ) ) ) ).
% calculation(1)
thf(fact_16_calculation_I2_J,axiom,
( ( ( uSubst95898978stappf @ sigma @ ua @ ( pred @ p @ theta ) )
!= ( uSubst95898978stappf @ ( uSubst969145931substt @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) @ ( bot_bot @ ( set @ variable ) )
@ ( the @ fml
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) )
=> ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p )
= ( none @ fml ) ) ) ).
% calculation(2)
thf(fact_17_f6,axiom,
( ( uSubst95898978stappf @ sigma @ ua @ ( pred @ p @ theta ) )
= ( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ p @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) )
@ ^ [F2: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ F2 ) @ ua )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) @ ( bot_bot @ ( set @ variable ) ) @ F2 )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) ).
% f6
thf(fact_18__092_060open_062usubstappf_A_092_060sigma_062_AU_A_IPred_Ap_A_092_060theta_062_J_A_092_060noteq_062_A_Icase_ASPreds_A_092_060sigma_062_Ap_Aof_ANone_A_092_060Rightarrow_062_AAfml_A_IPred_Ap_A_Ithe_A_Iusubstappt_A_092_060sigma_062_AV_A_092_060theta_062_J_J_J_A_124_ASome_Af_A_092_060Rightarrow_062_Aif_AFVF_Af_A_092_060inter_062_AV_A_061_A_123_125_Athen_Ausubstappf_A_Idotsubstt_A_Ithe_A_Iusubstappt_A_092_060sigma_062_AV_A_092_060theta_062_J_J_J_A_123_125_Af_Aelse_Aundeff_J_092_060close_062,axiom,
( ( uSubst95898978stappf @ sigma @ ua @ ( pred @ p @ theta ) )
!= ( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ p @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) )
@ ^ [F2: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ F2 ) @ va )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) @ ( bot_bot @ ( set @ variable ) ) @ F2 )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) ).
% \<open>usubstappf \<sigma> U (Pred p \<theta>) \<noteq> (case SPreds \<sigma> p of None \<Rightarrow> Afml (Pred p (the (usubstappt \<sigma> V \<theta>))) | Some f \<Rightarrow> if FVF f \<inter> V = {} then usubstappf (dotsubstt (the (usubstappt \<sigma> V \<theta>))) {} f else undeff)\<close>
thf(fact_19_usubstappf__pred2,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),P3: char,R2: fml,U: set @ variable,Theta: trm] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ Sigma
@ P3 )
= ( some @ fml @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ R2 ) @ U )
!= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst95898978stappf @ Sigma @ U @ ( pred @ P3 @ Theta ) )
= ( none @ fml ) ) ) ) ).
% usubstappf_pred2
thf(fact_20_f3,axiom,
( ( case_option @ ( option @ fml ) @ trm @ ( none @ fml )
@ ^ [T2: trm] :
( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ p @ T2 ) )
@ ^ [F2: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ F2 ) @ ua )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ T2 ) @ ( bot_bot @ ( set @ variable ) ) @ F2 )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) )
@ ( uSubst95898992stappt @ sigma @ ua @ theta ) )
= ( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ p @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ ua @ theta ) ) ) )
@ ^ [F2: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ F2 ) @ ua )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ ua @ theta ) ) ) @ ( bot_bot @ ( set @ variable ) ) @ F2 )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) ).
% f3
thf(fact_21_f8,axiom,
( ( case_option @ ( option @ fml ) @ trm @ ( none @ fml )
@ ^ [T2: trm] :
( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ p @ T2 ) )
@ ^ [F2: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ F2 ) @ va )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ T2 ) @ ( bot_bot @ ( set @ variable ) ) @ F2 )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) )
@ ( uSubst95898992stappt @ sigma @ va @ theta ) )
= ( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ p @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) )
@ ^ [F2: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ F2 ) @ va )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ ( the @ trm @ ( uSubst95898992stappt @ sigma @ va @ theta ) ) ) @ ( bot_bot @ ( set @ variable ) ) @ F2 )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) ) ) ).
% f8
thf(fact_22_f1,axiom,
( ( case_option @ ( option @ fml ) @ trm @ ( none @ fml )
@ ^ [T2: trm] :
( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ p @ T2 ) )
@ ^ [F2: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ F2 ) @ ua )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ T2 ) @ ( bot_bot @ ( set @ variable ) ) @ F2 )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) )
@ ( uSubst95898992stappt @ sigma @ ua @ theta ) )
!= ( none @ fml ) ) ).
% f1
thf(fact_23_f4,axiom,
( ( case_option @ ( option @ fml ) @ trm @ ( none @ fml )
@ ^ [T2: trm] :
( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ p @ T2 ) )
@ ^ [F2: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ F2 ) @ va )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ T2 ) @ ( bot_bot @ ( set @ variable ) ) @ F2 )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p ) )
@ ( uSubst95898992stappt @ sigma @ va @ theta ) )
!= ( none @ fml ) ) ).
% f4
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,X2: option @ A] :
( ( ! [Y: A] :
( X2
!= ( some @ A @ Y ) ) )
= ( X2
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_26_not__None__eq,axiom,
! [A: $tType,X2: option @ A] :
( ( X2
!= ( none @ A ) )
= ( ? [Y: A] :
( X2
= ( some @ A @ Y ) ) ) ) ).
% not_None_eq
thf(fact_27_usubstappf_Osimps_I1_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,P3: char,Theta: trm] :
( ( uSubst95898978stappf @ Sigma @ U @ ( pred @ P3 @ Theta ) )
= ( case_option @ ( option @ fml ) @ trm @ ( none @ fml )
@ ^ [Sigma_theta: trm] :
( case_option @ ( option @ fml ) @ fml @ ( some @ fml @ ( pred @ P3 @ Sigma_theta ) )
@ ^ [R: fml] :
( if @ ( option @ fml )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ R ) @ U )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898978stappf @ ( uSubst969145931substt @ Sigma_theta ) @ ( bot_bot @ ( set @ variable ) ) @ R )
@ ( none @ fml ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ Sigma
@ P3 ) )
@ ( uSubst95898992stappt @ Sigma @ U @ Theta ) ) ) ).
% usubstappf.simps(1)
thf(fact_28_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X2: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X2 )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_29_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X2: A] :
( ( inf_inf @ A @ X2 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_30_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P2: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P2 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P2 @ F1 ) )
| ( ( Option
= ( some @ A @ ( the @ A @ Option ) ) )
& ~ ( P2 @ ( F22 @ ( the @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_31_f2,axiom,
! [Z: option @ fml,F: trm > ( option @ fml ),Za: option @ trm] :
( ( ( Za
= ( none @ trm ) )
=> ( ( case_option @ ( option @ fml ) @ trm @ Z @ F @ Za )
= Z ) )
& ( ( Za
!= ( none @ trm ) )
=> ( ( case_option @ ( option @ fml ) @ trm @ Z @ F @ Za )
= ( F @ ( the @ trm @ Za ) ) ) ) ) ).
% f2
thf(fact_32_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ Y2 )
= ( inf_inf @ A @ X2 @ Y2 ) ) ) ).
% inf_right_idem
thf(fact_33_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_34_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y2: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y2 ) )
= ( inf_inf @ A @ X2 @ Y2 ) ) ) ).
% inf_left_idem
thf(fact_35_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_36_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A] :
( ( inf_inf @ A @ X2 @ X2 )
= X2 ) ) ).
% inf_idem
thf(fact_37_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A] :
( ( inf_inf @ A @ A2 @ A2 )
= A2 ) ) ).
% inf.idem
thf(fact_38_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_39_option_Oinject,axiom,
! [A: $tType,X23: A,Y22: A] :
( ( ( some @ A @ X23 )
= ( some @ A @ Y22 ) )
= ( X23 = Y22 ) ) ).
% option.inject
thf(fact_40_usubstappf__pred,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),P3: char,R2: fml,U: set @ variable,Theta: trm,Sigma_theta2: trm] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ Sigma
@ P3 )
= ( some @ fml @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ R2 ) @ U )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
= ( some @ trm @ Sigma_theta2 ) )
=> ( ( uSubst95898978stappf @ Sigma @ U @ ( pred @ P3 @ Theta ) )
= ( uSubst95898978stappf @ ( uSubst969145931substt @ Sigma_theta2 ) @ ( bot_bot @ ( set @ variable ) ) @ R2 ) ) ) ) ) ).
% usubstappf_pred
thf(fact_41_Diamondo_Oinduct,axiom,
! [P2: ( option @ game ) > ( option @ fml ) > $o,A0: option @ game,A1: option @ fml] :
( ! [Alpha: game,Phi: fml] : ( P2 @ ( some @ game @ Alpha ) @ ( some @ fml @ Phi ) )
=> ( ! [X_1: option @ fml] : ( P2 @ ( none @ game ) @ X_1 )
=> ( ! [V: game] : ( P2 @ ( some @ game @ V ) @ ( none @ fml ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% Diamondo.induct
thf(fact_42_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_43_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_44_usubstappt__det,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,V2: set @ variable] :
( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ V2 @ Theta )
!= ( none @ trm ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
= ( uSubst95898992stappt @ Sigma @ V2 @ Theta ) ) ) ) ).
% usubstappt_det
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G2: A > B] :
( ! [X3: A] :
( ( F3 @ X3 )
= ( G2 @ X3 ) )
=> ( F3 = G2 ) ) ).
% ext
thf(fact_49_case__optionE,axiom,
! [A: $tType,P2: $o,Q: A > $o,X2: option @ A] :
( ( case_option @ $o @ A @ P2 @ Q @ X2 )
=> ( ( ( X2
= ( none @ A ) )
=> ~ P2 )
=> ~ ! [Y3: A] :
( ( X2
= ( some @ A @ Y3 ) )
=> ~ ( Q @ Y3 ) ) ) ) ).
% case_optionE
thf(fact_50_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z2 ) )
= ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X2 @ Z2 ) ) ) ) ).
% inf_left_commute
thf(fact_51_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_52_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X: A,Y: A] : ( inf_inf @ A @ Y @ X ) ) ) ) ).
% inf_commute
thf(fact_53_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A4: A,B3: A] : ( inf_inf @ A @ B3 @ A4 ) ) ) ) ).
% inf.commute
thf(fact_54_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ Z2 )
= ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z2 ) ) ) ) ).
% inf_assoc
thf(fact_55_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_56_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_57_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_58_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_59_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_60_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y2 ) @ Z2 )
= ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z2 ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_61_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y2 @ Z2 ) )
= ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X2 @ Z2 ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_62_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y2: A] :
( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y2 ) )
= ( inf_inf @ A @ X2 @ Y2 ) ) ) ).
% inf_sup_aci(4)
thf(fact_63_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_64_option_Odistinct_I1_J,axiom,
! [A: $tType,X23: A] :
( ( none @ A )
!= ( some @ A @ X23 ) ) ).
% option.distinct(1)
thf(fact_65_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X23: A] :
( ( Option
= ( some @ A @ X23 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_66_option_Oexhaust,axiom,
! [A: $tType,Y2: option @ A] :
( ( Y2
!= ( none @ A ) )
=> ~ ! [X24: A] :
( Y2
!= ( some @ A @ X24 ) ) ) ).
% option.exhaust
thf(fact_67_option_Oinducts,axiom,
! [A: $tType,P2: ( option @ A ) > $o,Option: option @ A] :
( ( P2 @ ( none @ A ) )
=> ( ! [X3: A] : ( P2 @ ( some @ A @ X3 ) )
=> ( P2 @ Option ) ) ) ).
% option.inducts
thf(fact_68_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P4: ( option @ A ) > $o] :
? [X4: option @ A] : ( P4 @ X4 ) )
= ( ^ [P: ( option @ A ) > $o] :
( ( P @ ( none @ A ) )
| ? [X: A] : ( P @ ( some @ A @ X ) ) ) ) ) ).
% split_option_ex
thf(fact_69_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P4: ( option @ A ) > $o] :
! [X4: option @ A] : ( P4 @ X4 ) )
= ( ^ [P: ( option @ A ) > $o] :
( ( P @ ( none @ A ) )
& ! [X: A] : ( P @ ( some @ A @ X ) ) ) ) ) ).
% split_option_all
thf(fact_70_combine__options__cases,axiom,
! [A: $tType,B: $tType,X2: option @ A,P2: ( option @ A ) > ( option @ B ) > $o,Y2: option @ B] :
( ( ( X2
= ( none @ A ) )
=> ( P2 @ X2 @ Y2 ) )
=> ( ( ( Y2
= ( none @ B ) )
=> ( P2 @ X2 @ Y2 ) )
=> ( ! [A5: A,B5: B] :
( ( X2
= ( some @ A @ A5 ) )
=> ( ( Y2
= ( some @ B @ B5 ) )
=> ( P2 @ X2 @ Y2 ) ) )
=> ( P2 @ X2 @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_71_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_72_option_Oexpand,axiom,
! [A: $tType,Option: option @ A,Option2: option @ A] :
( ( ( Option
= ( none @ A ) )
= ( Option2
= ( none @ A ) ) )
=> ( ( ( Option
!= ( none @ A ) )
=> ( ( Option2
!= ( none @ A ) )
=> ( ( the @ A @ Option )
= ( the @ A @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_73_option_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > B,X23: A] :
( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X23 ) )
= ( F22 @ X23 ) ) ).
% option.simps(5)
thf(fact_74_option_Osel,axiom,
! [A: $tType,X23: A] :
( ( the @ A @ ( some @ A @ X23 ) )
= X23 ) ).
% option.sel
thf(fact_75_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_76_option_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_option @ B @ A )
= ( ^ [F12: B,F23: A > B,Option3: option @ A] :
( if @ B
@ ( Option3
= ( none @ A ) )
@ F12
@ ( F23 @ ( the @ A @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_77_usubstappf__pred__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,P3: char,Theta: trm] :
( ( ( uSubst95898978stappf @ Sigma @ U @ ( pred @ P3 @ Theta ) )
!= ( none @ fml ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
& ( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ Sigma
@ P3 )
= ( none @ fml ) )
| ? [R3: fml] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ Sigma
@ P3 )
= ( some @ fml @ R3 ) )
& ( ( inf_inf @ ( set @ variable ) @ ( static_FVF @ R3 ) @ U )
= ( bot_bot @ ( set @ variable ) ) ) ) ) ) ) ).
% usubstappf_pred_conv
thf(fact_78_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P2: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P2 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P2 @ F1 ) )
& ( ( Option
= ( some @ A @ ( the @ A @ Option ) ) )
=> ( P2 @ ( F22 @ ( the @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_79_fml_Oinject_I1_J,axiom,
! [X11: char,X12: trm,Y11: char,Y12: trm] :
( ( ( pred @ X11 @ X12 )
= ( pred @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% fml.inject(1)
thf(fact_80_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_81_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_82_empty__Collect__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P2 ) )
= ( ! [X: A] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_83_Collect__empty__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: A] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_84_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_85_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_86_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_87_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_88_Aterm__Some,axiom,
( ( some @ trm )
= ( some @ trm ) ) ).
% Aterm_Some
thf(fact_89_Loopo_Ocases,axiom,
! [X2: option @ game] :
( ! [Alpha: game] :
( X2
!= ( some @ game @ Alpha ) )
=> ( X2
= ( none @ game ) ) ) ).
% Loopo.cases
thf(fact_90_ODEo_Oinduct,axiom,
! [P2: char > ( option @ trm ) > $o,A0: char,A1: option @ trm] :
( ! [X3: char,Theta2: trm] : ( P2 @ X3 @ ( some @ trm @ Theta2 ) )
=> ( ! [X3: char] : ( P2 @ X3 @ ( none @ trm ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% ODEo.induct
thf(fact_91_Loopo_Oinduct,axiom,
! [P2: ( option @ game ) > $o,A0: option @ game] :
( ! [Alpha: game] : ( P2 @ ( some @ game @ Alpha ) )
=> ( ( P2 @ ( none @ game ) )
=> ( P2 @ A0 ) ) ) ).
% Loopo.induct
thf(fact_92_undefg__equiv,axiom,
! [Alpha2: option @ game] :
( ( Alpha2
!= ( none @ game ) )
= ( ? [G: game] :
( Alpha2
= ( some @ game @ G ) ) ) ) ).
% undefg_equiv
thf(fact_93_undeft__equiv,axiom,
! [Theta: option @ trm] :
( ( Theta
!= ( none @ trm ) )
= ( ? [T2: trm] :
( Theta
= ( some @ trm @ T2 ) ) ) ) ).
% undeft_equiv
thf(fact_94_Timeso_Oinduct,axiom,
! [P2: ( option @ trm ) > ( option @ trm ) > $o,A0: option @ trm,A1: option @ trm] :
( ! [Theta2: trm,Eta: trm] : ( P2 @ ( some @ trm @ Theta2 ) @ ( some @ trm @ Eta ) )
=> ( ! [X_1: option @ trm] : ( P2 @ ( none @ trm ) @ X_1 )
=> ( ! [V: trm] : ( P2 @ ( some @ trm @ V ) @ ( none @ trm ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% Timeso.induct
thf(fact_95_Assigno_Oinduct,axiom,
! [P2: variable > ( option @ trm ) > $o,A0: variable,A1: option @ trm] :
( ! [X3: variable,Theta2: trm] : ( P2 @ X3 @ ( some @ trm @ Theta2 ) )
=> ( ! [X3: variable] : ( P2 @ X3 @ ( none @ trm ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% Assigno.induct
thf(fact_96_Composeo_Oinduct,axiom,
! [P2: ( option @ game ) > ( option @ game ) > $o,A0: option @ game,A1: option @ game] :
( ! [Alpha: game,Beta: game] : ( P2 @ ( some @ game @ Alpha ) @ ( some @ game @ Beta ) )
=> ( ! [Alpha: option @ game] : ( P2 @ Alpha @ ( none @ game ) )
=> ( ! [V: game] : ( P2 @ ( none @ game ) @ ( some @ game @ V ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% Composeo.induct
thf(fact_97_Differentialo_Ocases,axiom,
! [X2: option @ trm] :
( ! [Theta2: trm] :
( X2
!= ( some @ trm @ Theta2 ) )
=> ( X2
= ( none @ trm ) ) ) ).
% Differentialo.cases
thf(fact_98_Differentialo_Oinduct,axiom,
! [P2: ( option @ trm ) > $o,A0: option @ trm] :
( ! [Theta2: trm] : ( P2 @ ( some @ trm @ Theta2 ) )
=> ( ( P2 @ ( none @ trm ) )
=> ( P2 @ A0 ) ) ) ).
% Differentialo.induct
thf(fact_99_undeft__None,axiom,
( ( none @ trm )
= ( none @ trm ) ) ).
% undeft_None
thf(fact_100_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_101_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_102_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_103_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_104_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_105_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_106_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_107_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_108_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_109_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_110_Int__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Int_absorb
thf(fact_111_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_112_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_113_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_114_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X: A] : $false ) ) ).
% empty_def
thf(fact_115_Collect__conj__eq,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( P2 @ X )
& ( Q @ X ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_116_Int__Collect,axiom,
! [A: $tType,X2: A,A3: set @ A,P2: A > $o] :
( ( member @ A @ X2 @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P2 ) ) )
= ( ( member @ A @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ).
% Int_Collect
thf(fact_117_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_118_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_119_Int__emptyI,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ~ ( member @ A @ X3 @ B4 ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B4 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_120_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_121_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_122_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_123_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P2: $o,Q: A > $o,X2: option @ A,R4: B > $o,F3: B,G2: A > B] :
( ( case_option @ $o @ A @ P2 @ Q @ X2 )
=> ( ( P2
=> ( R4 @ F3 ) )
=> ( ! [Q2: A] :
( ( Q @ Q2 )
=> ( R4 @ ( G2 @ Q2 ) ) )
=> ( R4 @ ( case_option @ B @ A @ F3 @ G2 @ X2 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_124_usubstappf__geqr,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta2: trm] :
( ( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
!= ( none @ fml ) )
=> ( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
= ( some @ fml @ ( geq @ ( the @ trm @ ( uSubst95898992stappt @ Sigma @ U @ Theta ) ) @ ( the @ trm @ ( uSubst95898992stappt @ Sigma @ U @ Eta2 ) ) ) ) ) ) ).
% usubstappf_geqr
thf(fact_125_usubstappf__geq,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta2: trm] :
( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Eta2 )
!= ( none @ trm ) )
=> ( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
= ( some @ fml @ ( geq @ ( the @ trm @ ( uSubst95898992stappt @ Sigma @ U @ Theta ) ) @ ( the @ trm @ ( uSubst95898992stappt @ Sigma @ U @ Eta2 ) ) ) ) ) ) ) ).
% usubstappf_geq
thf(fact_126_Geqo_Osimps_I3_J,axiom,
! [V3: trm] :
( ( uSubst1556497037e_Geqo @ ( some @ trm @ V3 ) @ ( none @ trm ) )
= ( none @ fml ) ) ).
% Geqo.simps(3)
thf(fact_127_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_128_usubstappf__geq__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta2: trm] :
( ( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
!= ( none @ fml ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
& ( ( uSubst95898992stappt @ Sigma @ U @ Eta2 )
!= ( none @ trm ) ) ) ) ).
% usubstappf_geq_conv
thf(fact_129_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_130_inf1I,axiom,
! [A: $tType,A3: A > $o,X2: A,B4: A > $o] :
( ( A3 @ X2 )
=> ( ( B4 @ X2 )
=> ( inf_inf @ ( A > $o ) @ A3 @ B4 @ X2 ) ) ) ).
% inf1I
thf(fact_131_inf1E,axiom,
! [A: $tType,A3: A > $o,B4: A > $o,X2: A] :
( ( inf_inf @ ( A > $o ) @ A3 @ B4 @ X2 )
=> ~ ( ( A3 @ X2 )
=> ~ ( B4 @ X2 ) ) ) ).
% inf1E
thf(fact_132_inf1D1,axiom,
! [A: $tType,A3: A > $o,B4: A > $o,X2: A] :
( ( inf_inf @ ( A > $o ) @ A3 @ B4 @ X2 )
=> ( A3 @ X2 ) ) ).
% inf1D1
thf(fact_133_inf1D2,axiom,
! [A: $tType,A3: A > $o,B4: A > $o,X2: A] :
( ( inf_inf @ ( A > $o ) @ A3 @ B4 @ X2 )
=> ( B4 @ X2 ) ) ).
% inf1D2
thf(fact_134_fml_Odistinct_I1_J,axiom,
! [X11: char,X12: trm,X21: trm,X222: trm] :
( ( pred @ X11 @ X12 )
!= ( geq @ X21 @ X222 ) ) ).
% fml.distinct(1)
thf(fact_135_usubstappf_Osimps_I2_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta2: trm] :
( ( uSubst95898978stappf @ Sigma @ U @ ( geq @ Theta @ Eta2 ) )
= ( uSubst1556497037e_Geqo @ ( uSubst95898992stappt @ Sigma @ U @ Theta ) @ ( uSubst95898992stappt @ Sigma @ U @ Eta2 ) ) ) ).
% usubstappf.simps(2)
thf(fact_136_Geqo_Osimps_I1_J,axiom,
! [Theta: trm,Eta2: trm] :
( ( uSubst1556497037e_Geqo @ ( some @ trm @ Theta ) @ ( some @ trm @ Eta2 ) )
= ( some @ fml @ ( geq @ Theta @ Eta2 ) ) ) ).
% Geqo.simps(1)
thf(fact_137_Geqo_Oelims,axiom,
! [X2: option @ trm,Xa: option @ trm,Y2: option @ fml] :
( ( ( uSubst1556497037e_Geqo @ X2 @ Xa )
= Y2 )
=> ( ! [Theta2: trm] :
( ( X2
= ( some @ trm @ Theta2 ) )
=> ! [Eta: trm] :
( ( Xa
= ( some @ trm @ Eta ) )
=> ( Y2
!= ( some @ fml @ ( geq @ Theta2 @ Eta ) ) ) ) )
=> ( ( ( X2
= ( none @ trm ) )
=> ( Y2
!= ( none @ fml ) ) )
=> ~ ( ? [V: trm] :
( X2
= ( some @ trm @ V ) )
=> ( ( Xa
= ( none @ trm ) )
=> ( Y2
!= ( none @ fml ) ) ) ) ) ) ) ).
% Geqo.elims
thf(fact_138_Geqo_Osimps_I2_J,axiom,
! [Eta2: option @ trm] :
( ( uSubst1556497037e_Geqo @ ( none @ trm ) @ Eta2 )
= ( none @ fml ) ) ).
% Geqo.simps(2)
thf(fact_139_Geqo__undef,axiom,
! [Theta: option @ trm,Eta2: option @ trm] :
( ( ( uSubst1556497037e_Geqo @ Theta @ Eta2 )
= ( none @ fml ) )
= ( ( Theta
= ( none @ trm ) )
| ( Eta2
= ( none @ trm ) ) ) ) ).
% Geqo_undef
thf(fact_140_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_141_Collect__empty__eq__bot,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( P2
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_142_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_143_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_144_usubstappt_Osimps_I4_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,F3: char,Theta: trm] :
( ( uSubst95898992stappt @ Sigma @ U @ ( func @ F3 @ Theta ) )
= ( case_option @ ( option @ trm ) @ trm @ ( none @ trm )
@ ^ [Sigma_theta: trm] :
( case_option @ ( option @ trm ) @ trm @ ( some @ trm @ ( func @ F3 @ Sigma_theta ) )
@ ^ [R: trm] :
( if @ ( option @ trm )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R ) @ U )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898992stappt @ ( uSubst969145931substt @ Sigma_theta ) @ ( bot_bot @ ( set @ variable ) ) @ R )
@ ( none @ trm ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F4: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F4 ) )
@ Sigma
@ F3 ) )
@ ( uSubst95898992stappt @ Sigma @ U @ Theta ) ) ) ).
% usubstappt.simps(4)
thf(fact_145_Diamondo_Ocases,axiom,
! [X2: product_prod @ ( option @ game ) @ ( option @ fml )] :
( ! [Alpha: game,Phi: fml] :
( X2
!= ( product_Pair @ ( option @ game ) @ ( option @ fml ) @ ( some @ game @ Alpha ) @ ( some @ fml @ Phi ) ) )
=> ( ! [Phi: option @ fml] :
( X2
!= ( product_Pair @ ( option @ game ) @ ( option @ fml ) @ ( none @ game ) @ Phi ) )
=> ~ ! [V: game] :
( X2
!= ( product_Pair @ ( option @ game ) @ ( option @ fml ) @ ( some @ game @ V ) @ ( none @ fml ) ) ) ) ) ).
% Diamondo.cases
thf(fact_146_Testo__undef,axiom,
! [Phi2: option @ fml] :
( ( ( uSubst190403692_Testo @ Phi2 )
= ( none @ game ) )
= ( Phi2
= ( none @ fml ) ) ) ).
% Testo_undef
thf(fact_147_prod_Oinject,axiom,
! [A: $tType,B: $tType,X13: A,X23: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X13 @ X23 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X13 = Y1 )
& ( X23 = Y22 ) ) ) ).
% prod.inject
thf(fact_148_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_149_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_150_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_151_case__prodI2,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,C2: A > B > $o] :
( ! [A5: A,B5: B] :
( ( P3
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( C2 @ A5 @ B5 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P3 ) ) ).
% case_prodI2
thf(fact_152_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P3: product_prod @ A @ B,C2: A > B > C > $o,X2: C] :
( ! [A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A5 @ B5 )
= P3 )
=> ( C2 @ A5 @ B5 @ X2 ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P3 @ X2 ) ) ).
% case_prodI2'
thf(fact_153_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_154_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P3: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
( ! [A5: A,B5: B] :
( ( P3
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( member @ C @ Z2 @ ( C2 @ A5 @ B5 ) ) )
=> ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P3 ) ) ) ).
% mem_case_prodI2
thf(fact_155_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_156_usubstappt__func2,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),F3: char,R2: trm,U: set @ variable,Theta: trm] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F4: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F4 ) )
@ Sigma
@ F3 )
= ( some @ trm @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R2 ) @ U )
!= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( func @ F3 @ Theta ) )
= ( none @ trm ) ) ) ) ).
% usubstappt_func2
thf(fact_157_usubstappt__func,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),F3: char,R2: trm,U: set @ variable,Theta: trm,Sigma_theta2: trm] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F4: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F4 ) )
@ Sigma
@ F3 )
= ( some @ trm @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R2 ) @ U )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
= ( some @ trm @ Sigma_theta2 ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( func @ F3 @ Theta ) )
= ( uSubst95898992stappt @ ( uSubst969145931substt @ Sigma_theta2 ) @ ( bot_bot @ ( set @ variable ) ) @ R2 ) ) ) ) ) ).
% usubstappt_func
thf(fact_158_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_159_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_160_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_161_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C,G2: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y3: B] :
( ( F3 @ X3 @ Y3 )
= ( G2 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F3 )
= G2 ) ) ).
% cond_case_prod_eta
thf(fact_162_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_163_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P2: B > C > A,Z2: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P2 @ Z2 ) )
=> ~ ! [X3: B,Y3: C] :
( ( Z2
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_164_surj__pair,axiom,
! [A: $tType,B: $tType,P3: product_prod @ A @ B] :
? [X3: A,Y3: B] :
( P3
= ( product_Pair @ A @ B @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_165_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P3: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P2 @ P3 ) ) ).
% prod_cases
thf(fact_166_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_167_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_168_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_169_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_170_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F5: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) )] :
~ ! [A5: A,B5: B,C4: C,D2: D,E2: E,F6: F5] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F5 ) @ D2 @ ( product_Pair @ E @ F5 @ E2 @ F6 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_171_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F5: $tType,G3: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
~ ! [A5: A,B5: B,C4: C,D2: D,E2: E,F6: F5,G4: G3] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E2 @ ( product_Pair @ F5 @ G3 @ F6 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_172_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B5: B,C4: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B5 @ C4 ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct3
thf(fact_173_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B5: B,C4: C,D2: D] : ( P2 @ ( 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 ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct4
thf(fact_174_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B5: B,C4: C,D2: D,E2: E] : ( P2 @ ( 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 ) ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct5
thf(fact_175_prod__induct6,axiom,
! [F5: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) )] :
( ! [A5: A,B5: B,C4: C,D2: D,E2: E,F6: F5] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F5 ) @ D2 @ ( product_Pair @ E @ F5 @ E2 @ F6 ) ) ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct6
thf(fact_176_prod__induct7,axiom,
! [G3: $tType,F5: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
( ! [A5: A,B5: B,C4: C,D2: D,E2: E,F6: F5,G4: G3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E2 @ ( product_Pair @ F5 @ G3 @ F6 @ G4 ) ) ) ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct7
thf(fact_177_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P3: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P3 ) )
=> ~ ! [X3: B,Y3: C] :
( ( P3
= ( product_Pair @ B @ C @ X3 @ Y3 ) )
=> ~ ( member @ A @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_178_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_179_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_180_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_181_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P3: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P3 )
=> ~ ! [X3: A,Y3: B] :
( ( P3
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 ) ) ) ).
% case_prodE
thf(fact_182_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_183_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P3: product_prod @ A @ B,Z2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P3 @ Z2 )
=> ~ ! [X3: A,Y3: B] :
( ( P3
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ~ ( C2 @ X3 @ Y3 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_184_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P3 )
= P3 ) ).
% case_prod_Pair_iden
thf(fact_185_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > B > C,X13: A,X23: B] :
( ( product_case_prod @ A @ B @ C @ F3 @ ( product_Pair @ A @ B @ X13 @ X23 ) )
= ( F3 @ X13 @ X23 ) ) ).
% old.prod.case
thf(fact_186_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_187_usubstappt__func__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,F3: char,Theta: trm] :
( ( ( uSubst95898992stappt @ Sigma @ U @ ( func @ F3 @ Theta ) )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
& ( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F4: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F4 ) )
@ Sigma
@ 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 ) )
@ Sigma
@ F3 )
= ( some @ trm @ R3 ) )
& ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R3 ) @ U )
= ( bot_bot @ ( set @ variable ) ) ) ) ) ) ) ).
% usubstappt_func_conv
thf(fact_188_Testo_Osimps_I2_J,axiom,
( ( uSubst190403692_Testo @ ( none @ fml ) )
= ( none @ game ) ) ).
% Testo.simps(2)
thf(fact_189_usappconst__simp,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),F3: char,R2: trm,U: set @ variable] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F0: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F0 ) )
@ Sigma
@ F3 )
= ( some @ trm @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R2 ) @ U )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst1138577137pconst @ Sigma @ U @ F3 )
= ( some @ trm @ R2 ) ) ) ) ).
% usappconst_simp
thf(fact_190_usappconst__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,F3: char] :
( ( ( uSubst1138577137pconst @ Sigma @ U @ 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 ) )
@ Sigma
@ 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 ) )
@ Sigma
@ F3 )
= ( some @ trm @ R3 ) )
& ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R3 ) @ U )
= ( bot_bot @ ( set @ variable ) ) ) ) ) ) ).
% usappconst_conv
thf(fact_191_usubstappt__const,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),F3: char,R2: trm,U: set @ variable] :
( ( ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F0: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F0 ) )
@ Sigma
@ F3 )
= ( some @ trm @ R2 ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R2 ) @ U )
= ( bot_bot @ ( set @ variable ) ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( const @ F3 ) )
= ( some @ trm @ R2 ) ) ) ) ).
% usubstappt_const
thf(fact_192_trm_Oinject_I3_J,axiom,
! [X32: char,Y32: char] :
( ( ( const @ X32 )
= ( const @ Y32 ) )
= ( X32 = Y32 ) ) ).
% trm.inject(3)
thf(fact_193_trm_Odistinct_I23_J,axiom,
! [X32: char,X41: char,X42: trm] :
( ( const @ X32 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(23)
thf(fact_194_usubstappt_Osimps_I3_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,F3: char] :
( ( uSubst95898992stappt @ Sigma @ U @ ( const @ F3 ) )
= ( uSubst1138577137pconst @ Sigma @ U @ F3 ) ) ).
% usubstappt.simps(3)
thf(fact_195_ODEo_Ocases,axiom,
! [X2: product_prod @ char @ ( option @ trm )] :
( ! [X3: char,Theta2: trm] :
( X2
!= ( product_Pair @ char @ ( option @ trm ) @ X3 @ ( some @ trm @ Theta2 ) ) )
=> ~ ! [X3: char] :
( X2
!= ( product_Pair @ char @ ( option @ trm ) @ X3 @ ( none @ trm ) ) ) ) ).
% ODEo.cases
thf(fact_196_Assigno_Ocases,axiom,
! [X2: product_prod @ variable @ ( option @ trm )] :
( ! [X3: variable,Theta2: trm] :
( X2
!= ( product_Pair @ variable @ ( option @ trm ) @ X3 @ ( some @ trm @ Theta2 ) ) )
=> ~ ! [X3: variable] :
( X2
!= ( product_Pair @ variable @ ( option @ trm ) @ X3 @ ( none @ trm ) ) ) ) ).
% Assigno.cases
thf(fact_197_Timeso_Ocases,axiom,
! [X2: product_prod @ ( option @ trm ) @ ( option @ trm )] :
( ! [Theta2: trm,Eta: trm] :
( X2
!= ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ Theta2 ) @ ( some @ trm @ Eta ) ) )
=> ( ! [Eta: option @ trm] :
( X2
!= ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( none @ trm ) @ Eta ) )
=> ~ ! [V: trm] :
( X2
!= ( product_Pair @ ( option @ trm ) @ ( option @ trm ) @ ( some @ trm @ V ) @ ( none @ trm ) ) ) ) ) ).
% Timeso.cases
thf(fact_198_Existso_Ocases,axiom,
! [X2: product_prod @ variable @ ( option @ fml )] :
( ! [X3: variable,Phi: fml] :
( X2
!= ( product_Pair @ variable @ ( option @ fml ) @ X3 @ ( some @ fml @ Phi ) ) )
=> ~ ! [X3: variable] :
( X2
!= ( product_Pair @ variable @ ( option @ fml ) @ X3 @ ( none @ fml ) ) ) ) ).
% Existso.cases
thf(fact_199_Ando_Ocases,axiom,
! [X2: product_prod @ ( option @ fml ) @ ( option @ fml )] :
( ! [Phi: fml,Psi: fml] :
( X2
!= ( product_Pair @ ( option @ fml ) @ ( option @ fml ) @ ( some @ fml @ Phi ) @ ( some @ fml @ Psi ) ) )
=> ( ! [Psi: option @ fml] :
( X2
!= ( product_Pair @ ( option @ fml ) @ ( option @ fml ) @ ( none @ fml ) @ Psi ) )
=> ~ ! [V: fml] :
( X2
!= ( product_Pair @ ( option @ fml ) @ ( option @ fml ) @ ( some @ fml @ V ) @ ( none @ fml ) ) ) ) ) ).
% Ando.cases
thf(fact_200_Composeo_Ocases,axiom,
! [X2: product_prod @ ( option @ game ) @ ( option @ game )] :
( ! [Alpha: game,Beta: game] :
( X2
!= ( product_Pair @ ( option @ game ) @ ( option @ game ) @ ( some @ game @ Alpha ) @ ( some @ game @ Beta ) ) )
=> ( ! [Alpha: option @ game] :
( X2
!= ( product_Pair @ ( option @ game ) @ ( option @ game ) @ Alpha @ ( none @ game ) ) )
=> ~ ! [V: game] :
( X2
!= ( product_Pair @ ( option @ game ) @ ( option @ game ) @ ( none @ game ) @ ( some @ game @ V ) ) ) ) ) ).
% Composeo.cases
thf(fact_201_usappconst__def,axiom,
( uSubst1138577137pconst
= ( ^ [Sigma2: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U2: set @ variable,F2: char] :
( case_option @ ( option @ trm ) @ trm @ ( some @ trm @ ( const @ F2 ) )
@ ^ [R: trm] :
( if @ ( option @ trm )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R ) @ U2 )
= ( bot_bot @ ( set @ variable ) ) )
@ ( some @ trm @ R )
@ ( none @ trm ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F0: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F0 ) )
@ Sigma2
@ F2 ) ) ) ) ).
% usappconst_def
thf(fact_202_usubstappt__const__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,F3: char] :
( ( ( uSubst95898992stappt @ Sigma @ U @ ( 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 ) )
@ Sigma
@ 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 ) )
@ Sigma
@ F3 )
= ( some @ trm @ R3 ) )
& ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R3 ) @ U )
= ( bot_bot @ ( set @ variable ) ) ) ) ) ) ).
% usubstappt_const_conv
thf(fact_203_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_204_usubstappt_Opsimps_I4_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,F3: 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 ) @ Sigma @ ( product_Pair @ ( set @ variable ) @ trm @ U @ ( func @ F3 @ Theta ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( func @ F3 @ Theta ) )
= ( case_option @ ( option @ trm ) @ trm @ ( none @ trm )
@ ^ [Sigma_theta: trm] :
( case_option @ ( option @ trm ) @ trm @ ( some @ trm @ ( func @ F3 @ Sigma_theta ) )
@ ^ [R: trm] :
( if @ ( option @ trm )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R ) @ U )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898992stappt @ ( uSubst969145931substt @ Sigma_theta ) @ ( bot_bot @ ( set @ variable ) ) @ R )
@ ( none @ trm ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F4: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F4 ) )
@ Sigma
@ F3 ) )
@ ( uSubst95898992stappt @ Sigma @ U @ Theta ) ) ) ) ).
% usubstappt.psimps(4)
thf(fact_205_usubstappt_Opsimps_I3_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,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 ) @ Sigma @ ( product_Pair @ ( set @ variable ) @ trm @ U @ ( const @ F3 ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( const @ F3 ) )
= ( uSubst1138577137pconst @ Sigma @ U @ F3 ) ) ) ).
% usubstappt.psimps(3)
thf(fact_206_usubstappt_Opsimps_I1_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,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 ) @ Sigma @ ( product_Pair @ ( set @ variable ) @ trm @ U @ ( var @ X2 ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( var @ X2 ) )
= ( some @ trm @ ( var @ X2 ) ) ) ) ).
% usubstappt.psimps(1)
thf(fact_207_usubstappt_Opsimps_I2_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,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 ) @ Sigma @ ( product_Pair @ ( set @ variable ) @ trm @ U @ ( number @ R2 ) ) ) )
=> ( ( uSubst95898992stappt @ Sigma @ U @ ( number @ R2 ) )
= ( some @ trm @ ( number @ R2 ) ) ) ) ).
% usubstappt.psimps(2)
thf(fact_208_trm_Oinject_I2_J,axiom,
! [X23: real,Y22: real] :
( ( ( number @ X23 )
= ( number @ Y22 ) )
= ( X23 = Y22 ) ) ).
% trm.inject(2)
thf(fact_209_trm_Oinject_I1_J,axiom,
! [X13: variable,Y1: variable] :
( ( ( var @ X13 )
= ( var @ Y1 ) )
= ( X13 = Y1 ) ) ).
% trm.inject(1)
thf(fact_210_trm_Odistinct_I15_J,axiom,
! [X23: real,X41: char,X42: trm] :
( ( number @ X23 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(15)
thf(fact_211_trm_Odistinct_I5_J,axiom,
! [X13: variable,X41: char,X42: trm] :
( ( var @ X13 )
!= ( func @ X41 @ X42 ) ) ).
% trm.distinct(5)
thf(fact_212_trm_Odistinct_I1_J,axiom,
! [X13: variable,X23: real] :
( ( var @ X13 )
!= ( number @ X23 ) ) ).
% trm.distinct(1)
thf(fact_213_trm_Odistinct_I13_J,axiom,
! [X23: real,X32: char] :
( ( number @ X23 )
!= ( const @ X32 ) ) ).
% trm.distinct(13)
thf(fact_214_trm_Odistinct_I3_J,axiom,
! [X13: variable,X32: char] :
( ( var @ X13 )
!= ( const @ X32 ) ) ).
% trm.distinct(3)
thf(fact_215_usubstappt_Osimps_I2_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,R2: real] :
( ( uSubst95898992stappt @ Sigma @ U @ ( number @ R2 ) )
= ( some @ trm @ ( number @ R2 ) ) ) ).
% usubstappt.simps(2)
thf(fact_216_usubstappt_Osimps_I1_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,X2: variable] :
( ( uSubst95898992stappt @ Sigma @ U @ ( var @ X2 ) )
= ( some @ trm @ ( var @ X2 ) ) ) ).
% usubstappt.simps(1)
thf(fact_217_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,P2: ( 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,X3: 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 @ X3 ) ) ) )
=> ( P2 @ Sigma3 @ U3 @ ( var @ X3 ) ) )
=> ( ! [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 ) ) ) )
=> ( P2 @ 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,F6: 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 @ F6 ) ) ) )
=> ( P2 @ Sigma3 @ U3 @ ( const @ F6 ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,F6: 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 ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( func @ F6 @ Theta2 ) ) ) )
=> ( ( P2 @ Sigma3 @ U3 @ Theta2 )
=> ( ! [X25: trm] :
( ( ( uSubst95898992stappt @ Sigma3 @ U3 @ Theta2 )
= ( 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
@ F6 )
= ( some @ trm @ X2a ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ X2a ) @ U3 )
= ( bot_bot @ ( set @ variable ) ) )
=> ( P2 @ ( uSubst969145931substt @ X25 ) @ ( bot_bot @ ( set @ variable ) ) @ X2a ) ) ) )
=> ( P2 @ Sigma3 @ U3 @ ( func @ F6 @ Theta2 ) ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta2: 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 @ Theta2 @ Eta ) ) ) )
=> ( ( P2 @ Sigma3 @ U3 @ Theta2 )
=> ( ( P2 @ Sigma3 @ U3 @ Eta )
=> ( P2 @ Sigma3 @ U3 @ ( plus @ Theta2 @ Eta ) ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta2: 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 @ Theta2 @ Eta ) ) ) )
=> ( ( P2 @ Sigma3 @ U3 @ Theta2 )
=> ( ( P2 @ Sigma3 @ U3 @ Eta )
=> ( P2 @ Sigma3 @ U3 @ ( times @ Theta2 @ Eta ) ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: 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 ) @ Sigma3 @ ( product_Pair @ ( set @ variable ) @ trm @ U3 @ ( differential @ Theta2 ) ) ) )
=> ( ( P2 @ Sigma3
@ ( collect @ variable
@ ^ [X: variable] : $true )
@ Theta2 )
=> ( P2 @ Sigma3 @ U3 @ ( differential @ Theta2 ) ) ) )
=> ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ) ) ) ) ).
% usubstappt.pinduct
thf(fact_218_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_219_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_220_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_221_trm_Oinject_I7_J,axiom,
! [X7: trm,Y7: trm] :
( ( ( differential @ X7 )
= ( differential @ Y7 ) )
= ( X7 = Y7 ) ) ).
% trm.inject(7)
thf(fact_222_trm_Odistinct_I7_J,axiom,
! [X13: variable,X51: trm,X52: trm] :
( ( var @ X13 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(7)
thf(fact_223_trm_Odistinct_I9_J,axiom,
! [X13: variable,X61: trm,X62: trm] :
( ( var @ X13 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(9)
thf(fact_224_trm_Odistinct_I11_J,axiom,
! [X13: variable,X7: trm] :
( ( var @ X13 )
!= ( differential @ X7 ) ) ).
% trm.distinct(11)
thf(fact_225_trm_Odistinct_I17_J,axiom,
! [X23: real,X51: trm,X52: trm] :
( ( number @ X23 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(17)
thf(fact_226_trm_Odistinct_I19_J,axiom,
! [X23: real,X61: trm,X62: trm] :
( ( number @ X23 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(19)
thf(fact_227_trm_Odistinct_I21_J,axiom,
! [X23: real,X7: trm] :
( ( number @ X23 )
!= ( differential @ X7 ) ) ).
% trm.distinct(21)
thf(fact_228_trm_Oexhaust,axiom,
! [Y2: trm] :
( ! [X14: variable] :
( Y2
!= ( var @ X14 ) )
=> ( ! [X24: real] :
( Y2
!= ( number @ X24 ) )
=> ( ! [X33: char] :
( Y2
!= ( const @ X33 ) )
=> ( ! [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_229_trm_Oinduct,axiom,
! [P2: trm > $o,Trm: trm] :
( ! [X3: variable] : ( P2 @ ( var @ X3 ) )
=> ( ! [X3: real] : ( P2 @ ( number @ X3 ) )
=> ( ! [X3: char] : ( P2 @ ( const @ X3 ) )
=> ( ! [X1a: char,X2a2: trm] :
( ( P2 @ X2a2 )
=> ( P2 @ ( func @ X1a @ X2a2 ) ) )
=> ( ! [X1a: trm,X2a2: trm] :
( ( P2 @ X1a )
=> ( ( P2 @ X2a2 )
=> ( P2 @ ( plus @ X1a @ X2a2 ) ) ) )
=> ( ! [X1a: trm,X2a2: trm] :
( ( P2 @ X1a )
=> ( ( P2 @ X2a2 )
=> ( P2 @ ( times @ X1a @ X2a2 ) ) ) )
=> ( ! [X3: trm] :
( ( P2 @ X3 )
=> ( P2 @ ( differential @ X3 ) ) )
=> ( P2 @ Trm ) ) ) ) ) ) ) ) ).
% trm.induct
thf(fact_230_trm_Odistinct_I25_J,axiom,
! [X32: char,X51: trm,X52: trm] :
( ( const @ X32 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(25)
thf(fact_231_trm_Odistinct_I27_J,axiom,
! [X32: char,X61: trm,X62: trm] :
( ( const @ X32 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(27)
thf(fact_232_trm_Odistinct_I29_J,axiom,
! [X32: char,X7: trm] :
( ( const @ X32 )
!= ( differential @ X7 ) ) ).
% trm.distinct(29)
thf(fact_233_trm_Odistinct_I41_J,axiom,
! [X61: trm,X62: trm,X7: trm] :
( ( times @ X61 @ X62 )
!= ( differential @ X7 ) ) ).
% trm.distinct(41)
thf(fact_234_trm_Odistinct_I39_J,axiom,
! [X51: trm,X52: trm,X7: trm] :
( ( plus @ X51 @ X52 )
!= ( differential @ X7 ) ) ).
% trm.distinct(39)
thf(fact_235_trm_Odistinct_I37_J,axiom,
! [X51: trm,X52: trm,X61: trm,X62: trm] :
( ( plus @ X51 @ X52 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(37)
thf(fact_236_trm_Odistinct_I31_J,axiom,
! [X41: char,X42: trm,X51: trm,X52: trm] :
( ( func @ X41 @ X42 )
!= ( plus @ X51 @ X52 ) ) ).
% trm.distinct(31)
thf(fact_237_trm_Odistinct_I33_J,axiom,
! [X41: char,X42: trm,X61: trm,X62: trm] :
( ( func @ X41 @ X42 )
!= ( times @ X61 @ X62 ) ) ).
% trm.distinct(33)
thf(fact_238_trm_Odistinct_I35_J,axiom,
! [X41: char,X42: trm,X7: trm] :
( ( func @ X41 @ X42 )
!= ( differential @ X7 ) ) ).
% trm.distinct(35)
thf(fact_239_usubstappt_Ocases,axiom,
! [X2: 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,X3: variable] :
( X2
!= ( 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 @ X3 ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,R3: real] :
( X2
!= ( 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,F6: char] :
( X2
!= ( 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 @ F6 ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,F6: char,Theta2: trm] :
( X2
!= ( 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 @ F6 @ Theta2 ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta2: trm,Eta: trm] :
( X2
!= ( 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 @ Theta2 @ Eta ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta2: trm,Eta: trm] :
( X2
!= ( 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 @ Theta2 @ Eta ) ) ) )
=> ~ ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta2: trm] :
( X2
!= ( 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 @ Theta2 ) ) ) ) ) ) ) ) ) ) ).
% usubstappt.cases
thf(fact_240_usubstappt__times__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta2: trm] :
( ( ( uSubst95898992stappt @ Sigma @ U @ ( times @ Theta @ Eta2 ) )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
& ( ( uSubst95898992stappt @ Sigma @ U @ Eta2 )
!= ( none @ trm ) ) ) ) ).
% usubstappt_times_conv
thf(fact_241_usubstappt__plus__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta2: trm] :
( ( ( uSubst95898992stappt @ Sigma @ U @ ( plus @ Theta @ Eta2 ) )
!= ( none @ trm ) )
=> ( ( ( uSubst95898992stappt @ Sigma @ U @ Theta )
!= ( none @ trm ) )
& ( ( uSubst95898992stappt @ Sigma @ U @ Eta2 )
!= ( none @ trm ) ) ) ) ).
% usubstappt_plus_conv
thf(fact_242_usubstappt__differential__conv,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm] :
( ( ( uSubst95898992stappt @ Sigma @ U @ ( differential @ Theta ) )
!= ( none @ trm ) )
=> ( ( uSubst95898992stappt @ Sigma
@ ( collect @ variable
@ ^ [X: variable] : $true )
@ Theta )
!= ( none @ trm ) ) ) ).
% usubstappt_differential_conv
thf(fact_243_usubstappt__induct,axiom,
! [P2: ( 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,X3: variable] : ( P2 @ Sigma3 @ U3 @ ( var @ X3 ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,R3: real] : ( P2 @ 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,F6: char] : ( P2 @ Sigma3 @ U3 @ ( const @ F6 ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,F6: char,Theta2: trm] :
( ( P2 @ Sigma3 @ U3 @ Theta2 )
=> ( ! [X25: trm] :
( ( ( uSubst95898992stappt @ Sigma3 @ U3 @ Theta2 )
= ( 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
@ F6 )
= ( some @ trm @ X2a ) )
=> ( ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ X2a ) @ U3 )
= ( bot_bot @ ( set @ variable ) ) )
=> ( P2 @ ( uSubst969145931substt @ X25 ) @ ( bot_bot @ ( set @ variable ) ) @ X2a ) ) ) )
=> ( P2 @ Sigma3 @ U3 @ ( func @ F6 @ Theta2 ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta2: trm] :
( ( P2 @ Sigma3 @ U3 @ Theta2 )
=> ! [Eta: trm] :
( ( P2 @ Sigma3 @ U3 @ Eta )
=> ( P2 @ Sigma3 @ U3 @ ( plus @ Theta2 @ Eta ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta2: trm] :
( ( P2 @ Sigma3 @ U3 @ Theta2 )
=> ! [Eta: trm] :
( ( P2 @ Sigma3 @ U3 @ Eta )
=> ( P2 @ Sigma3 @ U3 @ ( times @ Theta2 @ Eta ) ) ) )
=> ( ! [Sigma3: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U3: set @ variable,Theta2: trm] :
( ( P2 @ Sigma3
@ ( collect @ variable
@ ^ [X: variable] : $true )
@ Theta2 )
=> ( P2 @ Sigma3 @ U3 @ ( differential @ Theta2 ) ) )
=> ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ) ) ) ).
% usubstappt_induct
thf(fact_244_dot__def,axiom,
( uSubst_Mirabelle_dot
= ( const @ ( char2 @ $false @ $true @ $true @ $true @ $false @ $true @ $false @ $false ) ) ) ).
% dot_def
thf(fact_245_usubstappt_Opelims,axiom,
! [X2: 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 @ X2 @ 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 ) @ X2 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ Xb ) ) )
=> ( ! [X3: variable] :
( ( Xb
= ( var @ X3 ) )
=> ( ( Y2
= ( some @ trm @ ( var @ X3 ) ) )
=> ~ ( 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 ) @ X2 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ ( var @ X3 ) ) ) ) ) )
=> ( ! [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 ) @ X2 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ ( number @ R3 ) ) ) ) ) )
=> ( ! [F6: char] :
( ( Xb
= ( const @ F6 ) )
=> ( ( Y2
= ( uSubst1138577137pconst @ X2 @ Xa @ F6 ) )
=> ~ ( 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 ) @ X2 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ ( const @ F6 ) ) ) ) ) )
=> ( ! [F6: char,Theta2: trm] :
( ( Xb
= ( func @ F6 @ Theta2 ) )
=> ( ( Y2
= ( case_option @ ( option @ trm ) @ trm @ ( none @ trm )
@ ^ [Sigma_theta: trm] :
( case_option @ ( option @ trm ) @ trm @ ( some @ trm @ ( func @ F6 @ Sigma_theta ) )
@ ^ [R: trm] :
( if @ ( option @ trm )
@ ( ( inf_inf @ ( set @ variable ) @ ( static_FVT @ R ) @ Xa )
= ( bot_bot @ ( set @ variable ) ) )
@ ( uSubst95898992stappt @ ( uSubst969145931substt @ Sigma_theta ) @ ( bot_bot @ ( set @ variable ) ) @ R )
@ ( none @ trm ) )
@ ( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ) @ ( char > ( option @ trm ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ trm ) )
@ ^ [F4: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ trm ) )
@ ^ [Uv: char > ( option @ fml ),Uw: char > ( option @ game )] : F4 ) )
@ X2
@ F6 ) )
@ ( uSubst95898992stappt @ X2 @ Xa @ Theta2 ) ) )
=> ~ ( 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 ) @ X2 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ ( func @ F6 @ Theta2 ) ) ) ) ) )
=> ( ! [Theta2: trm,Eta: trm] :
( ( Xb
= ( plus @ Theta2 @ Eta ) )
=> ( ( Y2
= ( uSubst1112714340_Pluso @ ( uSubst95898992stappt @ X2 @ Xa @ Theta2 ) @ ( uSubst95898992stappt @ X2 @ 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 ) @ X2 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ ( plus @ Theta2 @ Eta ) ) ) ) ) )
=> ( ! [Theta2: trm,Eta: trm] :
( ( Xb
= ( times @ Theta2 @ Eta ) )
=> ( ( Y2
= ( uSubst277968634Timeso @ ( uSubst95898992stappt @ X2 @ Xa @ Theta2 ) @ ( uSubst95898992stappt @ X2 @ 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 ) @ X2 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ ( times @ Theta2 @ Eta ) ) ) ) ) )
=> ~ ! [Theta2: trm] :
( ( Xb
= ( differential @ Theta2 ) )
=> ( ( Y2
= ( uSubst259074819ntialo
@ ( uSubst95898992stappt @ X2
@ ( collect @ variable
@ ^ [X: variable] : $true )
@ Theta2 ) ) )
=> ~ ( 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 ) @ X2 @ ( product_Pair @ ( set @ variable ) @ trm @ Xa @ ( differential @ Theta2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% usubstappt.pelims
thf(fact_246_Differentialo_Osimps_I2_J,axiom,
( ( uSubst259074819ntialo @ ( none @ trm ) )
= ( none @ trm ) ) ).
% Differentialo.simps(2)
thf(fact_247_Differentialo__undef,axiom,
! [Theta: option @ trm] :
( ( ( uSubst259074819ntialo @ Theta )
= ( none @ trm ) )
= ( Theta
= ( none @ trm ) ) ) ).
% Differentialo_undef
thf(fact_248_Pluso_Osimps_I2_J,axiom,
! [Eta2: option @ trm] :
( ( uSubst1112714340_Pluso @ ( none @ trm ) @ Eta2 )
= ( none @ trm ) ) ).
% Pluso.simps(2)
thf(fact_249_Timeso_Osimps_I2_J,axiom,
! [Eta2: option @ trm] :
( ( uSubst277968634Timeso @ ( none @ trm ) @ Eta2 )
= ( none @ trm ) ) ).
% Timeso.simps(2)
thf(fact_250_Pluso__undef,axiom,
! [Theta: option @ trm,Eta2: option @ trm] :
( ( ( uSubst1112714340_Pluso @ Theta @ Eta2 )
= ( none @ trm ) )
= ( ( Theta
= ( none @ trm ) )
| ( Eta2
= ( none @ trm ) ) ) ) ).
% Pluso_undef
thf(fact_251_Timeso__undef,axiom,
! [Theta: option @ trm,Eta2: option @ trm] :
( ( ( uSubst277968634Timeso @ Theta @ Eta2 )
= ( none @ trm ) )
= ( ( Theta
= ( none @ trm ) )
| ( Eta2
= ( none @ trm ) ) ) ) ).
% Timeso_undef
thf(fact_252_Pluso_Osimps_I3_J,axiom,
! [V3: trm] :
( ( uSubst1112714340_Pluso @ ( some @ trm @ V3 ) @ ( none @ trm ) )
= ( none @ trm ) ) ).
% Pluso.simps(3)
thf(fact_253_Timeso_Osimps_I3_J,axiom,
! [V3: trm] :
( ( uSubst277968634Timeso @ ( some @ trm @ V3 ) @ ( none @ trm ) )
= ( none @ trm ) ) ).
% Timeso.simps(3)
thf(fact_254_usubstappt_Osimps_I5_J,axiom,
! [Sigma: product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) ),U: set @ variable,Theta: trm,Eta2: trm] :
( ( uSubst95898992stappt @ Sigma @ U @ ( plus @ Theta @ Eta2 ) )
= ( uSubst1112714340_Pluso @ ( uSubst95898992stappt @ Sigma @ U @ Theta ) @ ( uSubst95898992stappt @ Sigma @ U @ Eta2 ) ) ) ).
% usubstappt.simps(5)
% 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,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( if @ A @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( if @ A @ $true @ X2 @ Y2 )
= X2 ) ).
% 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 @ fml ) )
@ ^ [Uu: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ trm ) ) @ ( product_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) ) @ ( char > ( option @ fml ) )
@ ^ [Uv: char > ( option @ trm )] :
( product_case_prod @ ( char > ( option @ fml ) ) @ ( char > ( option @ game ) ) @ ( char > ( option @ fml ) )
@ ^ [P: char > ( option @ fml ),Uw: char > ( option @ game )] : P ) )
@ sigma
@ p )
= ( none @ fml ) ) ).
%------------------------------------------------------------------------------