TPTP Problem File: LCL809_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL809_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 167
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_167 [Bla13]
% Status : Theorem
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 170 ( 80 unt; 54 typ; 0 def)
% Number of atoms : 174 ( 115 equ)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 86 ( 28 ~; 8 |; 10 &)
% ( 17 <=>; 23 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 45 ( 26 >; 19 *; 0 +; 0 <<)
% Number of predicates : 7 ( 6 usr; 0 prp; 1-3 aty)
% Number of functors : 42 ( 42 usr; 15 con; 0-5 aty)
% Number of variables : 280 ( 243 !; 7 ?; 280 :)
% ( 30 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:22:09
%------------------------------------------------------------------------------
%----Should-be-implicit typings (8)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Lambda_OdB,type,
dB: $tType ).
tff(ty_tc_List_Olist,type,
list: $tType > $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_String_Ochar,type,
char1: $tType ).
tff(ty_tc_String_Oliteral,type,
literal: $tType ).
tff(ty_tc_String_Onibble,type,
nibble: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (46)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).
tff(sy_c_COMBI,type,
combi:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_HOL_Obool_Obool__size,type,
bool_size: bool > nat ).
tff(sy_c_InductTermi_OIT,type,
it: fun(dB,bool) ).
tff(sy_c_Lambda_OdB_OAbs,type,
abs: dB > dB ).
tff(sy_c_Lambda_OdB_OApp,type,
app: fun(dB,fun(dB,dB)) ).
tff(sy_c_Lambda_OdB_OVar,type,
var: nat > dB ).
tff(sy_c_Lambda_OdB_OdB__case,type,
dB_case:
!>[T1: $tType] : ( ( fun(nat,T1) * fun(dB,fun(dB,T1)) * fun(dB,T1) * dB ) > T1 ) ).
tff(sy_c_Lambda_OdB_OdB__size,type,
dB_size: dB > nat ).
tff(sy_c_Lambda_Olift,type,
lift: fun(dB,fun(nat,dB)) ).
tff(sy_c_Lambda_Oliftn,type,
liftn: ( nat * dB * nat ) > dB ).
tff(sy_c_Lambda_Osubst,type,
subst: fun(dB,fun(dB,fun(nat,dB))) ).
tff(sy_c_Lambda_Osubstn,type,
substn: ( dB * dB * nat ) > dB ).
tff(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(A,B)) * B * list(A) ) > B ) ).
tff(sy_c_List_Olistsp,type,
listsp:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Omap,type,
map:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > list(B) ) ).
tff(sy_c_List_Omonoid__add__class_Olistsum,type,
monoid_add_listsum:
!>[A: $tType] : ( list(A) > A ) ).
tff(sy_c_Nat_OSuc,type,
suc: nat > nat ).
tff(sy_c_Nat_Onat_Onat__size,type,
nat_size: nat > nat ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_String_Ochar_OChar,type,
char: ( nibble * nibble ) > char1 ).
tff(sy_c_String_Ochar_Ochar__case,type,
char_case:
!>[T1: $tType] : ( ( fun(nibble,fun(nibble,T1)) * char1 ) > T1 ) ).
tff(sy_c_String_Ochar_Ochar__rec,type,
char_rec:
!>[T1: $tType] : ( ( fun(nibble,fun(nibble,T1)) * char1 ) > T1 ) ).
tff(sy_c_String_Ochar_Ochar__size,type,
char_size: char1 > nat ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_a____,type,
a: dB ).
tff(sy_v_b____,type,
b: dB ).
tff(sy_v_bs____,type,
bs: list(dB) ).
tff(sy_v_i____,type,
i: nat ).
tff(sy_v_n____,type,
n: nat ).
tff(sy_v_t____,type,
t: dB ).
tff(sy_v_u____,type,
u: dB ).
tff(sy_v_ua______,type,
ua: dB ).
%----Relevant facts (98)
tff(fact_0__096IT_At_096,axiom,
pp(aa(dB,bool,it,t)) ).
tff(fact_1_Var_I3_J,axiom,
pp(aa(dB,bool,it,ua)) ).
tff(fact_2_uIT,axiom,
pp(aa(dB,bool,it,u)) ).
tff(fact_3__096IT_A_Ib_091u_Pi_093_J_096,axiom,
pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,b),u),i))) ).
tff(fact_4_True,axiom,
n = i ).
tff(fact_5_lift__IT,axiom,
! [I: nat,T: dB] :
( pp(aa(dB,bool,it,T))
=> pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),lift,T),I))) ) ).
tff(fact_6_subst__lift,axiom,
! [S2: dB,K1: nat,T: dB] : aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,aa(nat,dB,aa(dB,fun(nat,dB),lift,T),K1)),S2),K1) = T ).
tff(fact_7__096IT_A_Iu_A_092_060degree_062_Aa_091u_Pi_093_J_096,axiom,
pp(aa(dB,bool,it,aa(dB,dB,aa(dB,fun(dB,dB),app,u),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,a),u),i)))) ).
tff(fact_8__096IT_A_I_Ilift_Au_A0_A_092_060degree_062_AVar_A0_J_091a_091u_Pi_093_P0_093_J_096,axiom,
pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,aa(dB,dB,aa(dB,fun(dB,dB),app,aa(nat,dB,aa(dB,fun(nat,dB),lift,u),zero_zero(nat))),var(zero_zero(nat)))),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,a),u),i)),zero_zero(nat)))) ).
tff(fact_9_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X3: A] :
( ( zero_zero(A) = X3 )
<=> ( X3 = zero_zero(A) ) ) ) ).
tff(fact_10_subst__Var__IT,axiom,
! [J: nat,I: nat,R3: dB] :
( pp(aa(dB,bool,it,R3))
=> pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,R3),var(I)),J))) ) ).
tff(fact_11_bool_Osize_I1_J,axiom,
bool_size(fTrue) = zero_zero(nat) ).
tff(fact_12_bool_Osize_I2_J,axiom,
bool_size(fFalse) = zero_zero(nat) ).
tff(fact_13_char__size,axiom,
! [C2: char1] : char_size(C2) = zero_zero(nat) ).
tff(fact_14_substn__subst__0,axiom,
! [S2: dB,T: dB] : substn(T,S2,zero_zero(nat)) = aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T),S2),zero_zero(nat)) ).
tff(fact_15_Cons_I3_J,axiom,
listsp(dB,it,map(dB,dB,combc(dB,nat,dB,lift,zero_zero(nat)),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun(nat,dB),subst,u),i),bs))) ).
tff(fact_16_nat_Osize_I1_J,axiom,
nat_size(zero_zero(nat)) = zero_zero(nat) ).
tff(fact_17_dB_Osimps_I2_J,axiom,
! [DB23: dB,DB13: dB,DB21: dB,DB11: dB] :
( ( aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21) = aa(dB,dB,aa(dB,fun(dB,dB),app,DB13),DB23) )
<=> ( ( DB11 = DB13 )
& ( DB21 = DB23 ) ) ) ).
tff(fact_18_dB_Osimps_I1_J,axiom,
! [Nat5: nat,Nat4: nat] :
( ( var(Nat4) = var(Nat5) )
<=> ( Nat4 = Nat5 ) ) ).
tff(fact_19_subst__App,axiom,
! [K1: nat,S2: dB,U: dB,T: dB] : aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,aa(dB,dB,aa(dB,fun(dB,dB),app,T),U)),S2),K1) = aa(dB,dB,aa(dB,fun(dB,dB),app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T),S2),K1)),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,U),S2),K1)) ).
tff(fact_20_lift_Osimps_I2_J,axiom,
! [K1: nat,T: dB,S2: dB] : aa(nat,dB,aa(dB,fun(nat,dB),lift,aa(dB,dB,aa(dB,fun(dB,dB),app,S2),T)),K1) = aa(dB,dB,aa(dB,fun(dB,dB),app,aa(nat,dB,aa(dB,fun(nat,dB),lift,S2),K1)),aa(nat,dB,aa(dB,fun(nat,dB),lift,T),K1)) ).
tff(fact_21_substn_Osimps_I2_J,axiom,
! [K1: nat,S2: dB,U: dB,T: dB] : substn(aa(dB,dB,aa(dB,fun(dB,dB),app,T),U),S2,K1) = aa(dB,dB,aa(dB,fun(dB,dB),app,substn(T,S2,K1)),substn(U,S2,K1)) ).
tff(fact_22_dB_Osimps_I4_J,axiom,
! [DB22: dB,DB12: dB,Nat: nat] : var(Nat) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB12),DB22) ).
tff(fact_23_dB_Osimps_I5_J,axiom,
! [Nat: nat,DB22: dB,DB12: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB12),DB22) != var(Nat) ).
tff(fact_24_app__Var__IT,axiom,
! [I: nat,T: dB] :
( pp(aa(dB,bool,it,T))
=> pp(aa(dB,bool,it,aa(dB,dB,aa(dB,fun(dB,dB),app,T),var(I)))) ) ).
tff(fact_25_lifts__IT,axiom,
! [Ts: list(dB)] :
( listsp(dB,it,Ts)
=> listsp(dB,it,map(dB,dB,combc(dB,nat,dB,lift,zero_zero(nat)),Ts)) ) ).
tff(fact_26_subst__eq,axiom,
! [U: dB,K1: nat] : aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,var(K1)),U),K1) = U ).
tff(fact_27_Var__IT,axiom,
! [N: nat] : pp(aa(dB,bool,it,var(N))) ).
tff(fact_28_listsp__conj__eq,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),X2: list(A)] :
( listsp(A,combs(A,bool,bool,aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),A1),B1),X2)
<=> ( listsp(A,A1,X2)
& listsp(A,B1,X2) ) ) ).
tff(fact_29_map__ident,axiom,
! [A: $tType,X2: list(A)] : map(A,A,combi(A),X2) = X2 ).
tff(fact_30_substn__subst__n,axiom,
! [N: nat,S2: dB,T: dB] : substn(T,S2,N) = aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T),liftn(N,S2,zero_zero(nat))),N) ).
tff(fact_31_dB_Osize_I1_J,axiom,
! [Nat: nat] : dB_size(var(Nat)) = zero_zero(nat) ).
tff(fact_32_size__char,axiom,
! [C2: char1] : size_size(char1,C2) = zero_zero(nat) ).
tff(fact_33_bool_Osize_I4_J,axiom,
size_size(bool,fFalse) = zero_zero(nat) ).
tff(fact_34_bool_Osize_I3_J,axiom,
size_size(bool,fTrue) = zero_zero(nat) ).
tff(fact_35_dB_Osize_I4_J,axiom,
! [Nat: nat] : size_size(dB,var(Nat)) = zero_zero(nat) ).
tff(fact_36_liftn_Osimps_I2_J,axiom,
! [K1: nat,T: dB,S2: dB,N: nat] : liftn(N,aa(dB,dB,aa(dB,fun(dB,dB),app,S2),T),K1) = aa(dB,dB,aa(dB,fun(dB,dB),app,liftn(N,S2,K1)),liftn(N,T,K1)) ).
tff(fact_37_liftn__0,axiom,
! [K1: nat,T: dB] : liftn(zero_zero(nat),T,K1) = T ).
tff(fact_38_size__bool,axiom,
! [Ba: bool] : size_size(bool,Ba) = zero_zero(nat) ).
tff(fact_39_size__literal__def,axiom,
! [S2: literal] : size_size(literal,S2) = zero_zero(nat) ).
tff(fact_40_nat_Osize_I3_J,axiom,
size_size(nat,zero_zero(nat)) = zero_zero(nat) ).
tff(fact_41_dB_Osimps_I10_J,axiom,
! [A: $tType,Nat4: nat,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,var(Nat4)) = aa(nat,A,F1,Nat4) ).
tff(fact_42_dB_Osimps_I11_J,axiom,
! [A: $tType,DB21: dB,DB11: dB,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21)) = aa(dB,A,aa(dB,fun(dB,A),F2,DB11),DB21) ).
tff(fact_43_char_Osize_I2_J,axiom,
! [Nibble22: nibble,Nibble12: nibble] : size_size(char1,char(Nibble12,Nibble22)) = zero_zero(nat) ).
tff(fact_44_char_Oinject,axiom,
! [Nibble23: nibble,Nibble13: nibble,Nibble2: nibble,Nibble1: nibble] :
( ( char(Nibble1,Nibble2) = char(Nibble13,Nibble23) )
<=> ( ( Nibble1 = Nibble13 )
& ( Nibble2 = Nibble23 ) ) ) ).
tff(fact_45_char_Osize_I1_J,axiom,
! [Nibble22: nibble,Nibble12: nibble] : char_size(char(Nibble12,Nibble22)) = zero_zero(nat) ).
tff(fact_46_nat__size,axiom,
! [N: nat] : size_size(nat,N) = N ).
tff(fact_47_char_Oexhaust,axiom,
! [Y: char1] :
~ ! [Nibble11: nibble,Nibble21: nibble] : Y != char(Nibble11,Nibble21) ).
tff(fact_48_char_Osimps_I2_J,axiom,
! [A: $tType,Nibble2: nibble,Nibble1: nibble,F1: fun(nibble,fun(nibble,A))] : char_case(A,F1,char(Nibble1,Nibble2)) = aa(nibble,A,aa(nibble,fun(nibble,A),F1,Nibble1),Nibble2) ).
tff(fact_49_char_Orecs,axiom,
! [A: $tType,Nibble2: nibble,Nibble1: nibble,F1: fun(nibble,fun(nibble,A))] : char_rec(A,F1,char(Nibble1,Nibble2)) = aa(nibble,A,aa(nibble,fun(nibble,A),F1,Nibble1),Nibble2) ).
tff(fact_50_listsum__0,axiom,
! [B: $tType,A: $tType] :
( monoid_add(A)
=> ! [Xs: list(B)] : monoid_add_listsum(A,map(B,A,combk(A,B,zero_zero(A)),Xs)) = zero_zero(A) ) ).
tff(fact_51_IT_OVar,axiom,
! [Na: nat,Rsa: list(dB)] :
( listsp(dB,it,Rsa)
=> pp(aa(dB,bool,it,foldl(dB,dB,app,var(Na),Rsa))) ) ).
tff(fact_52_liftn__lift,axiom,
! [K1: nat,T: dB,N: nat] : liftn(suc(N),T,K1) = aa(nat,dB,aa(dB,fun(nat,dB),lift,liftn(N,T,K1)),K1) ).
tff(fact_53_lift__map,axiom,
! [Ib: nat,Ts: list(dB),Ta: dB] : aa(nat,dB,aa(dB,fun(nat,dB),lift,foldl(dB,dB,app,Ta,Ts)),Ib) = foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),lift,Ta),Ib),map(dB,dB,combc(dB,nat,dB,lift,Ib),Ts)) ).
tff(fact_54_subst__map,axiom,
! [Ib: nat,Ub: dB,Ts: list(dB),Ta: dB] : aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,foldl(dB,dB,app,Ta,Ts)),Ub),Ib) = foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,Ta),Ub),Ib),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun(nat,dB),subst,Ub),Ib),Ts)) ).
tff(fact_55_apps__eq__tail__conv,axiom,
! [S1: dB,Ts: list(dB),R2: dB] :
( ( foldl(dB,dB,app,R2,Ts) = foldl(dB,dB,app,S1,Ts) )
<=> ( R2 = S1 ) ) ).
tff(fact_56_Var__apps__eq__Var__apps__conv,axiom,
! [Ss1: list(dB),Na: nat,Rsa: list(dB),M1: nat] :
( ( foldl(dB,dB,app,var(M1),Rsa) = foldl(dB,dB,app,var(Na),Ss1) )
<=> ( ( M1 = Na )
& ( Rsa = Ss1 ) ) ) ).
tff(fact_57_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,Xs: list(C),F: fun(C,B),Aa: A,G: fun(A,fun(B,A))] : foldl(A,B,G,Aa,map(C,B,F,Xs)) = foldl(A,C,combc(A,fun(C,B),fun(C,A),aa(fun(A,fun(B,A)),fun(A,fun(fun(C,B),fun(C,A))),aa(fun(fun(B,A),fun(fun(C,B),fun(C,A))),fun(fun(A,fun(B,A)),fun(A,fun(fun(C,B),fun(C,A)))),combb(fun(B,A),fun(fun(C,B),fun(C,A)),A),combb(B,A,C)),G),F),Aa,Xs) ).
tff(fact_58_nat_Oinject,axiom,
! [Nat5: nat,Nat4: nat] :
( ( suc(Nat4) = suc(Nat5) )
<=> ( Nat4 = Nat5 ) ) ).
tff(fact_59_Beta,axiom,
! [Ss1: list(dB),S1: dB,R2: dB] :
( pp(aa(dB,bool,it,foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,R2),S1),zero_zero(nat)),Ss1)))
=> ( pp(aa(dB,bool,it,S1))
=> pp(aa(dB,bool,it,foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(R2)),S1),Ss1))) ) ) ).
tff(fact_60_dB_Osimps_I3_J,axiom,
! [DB5: dB,DB3: dB] :
( ( abs(DB3) = abs(DB5) )
<=> ( DB3 = DB5 ) ) ).
tff(fact_61_Lambda,axiom,
! [R3: dB] :
( pp(aa(dB,bool,it,R3))
=> pp(aa(dB,bool,it,abs(R3))) ) ).
tff(fact_62_Abs__apps__eq__Abs__apps__conv,axiom,
! [Ss1: list(dB),S1: dB,Rsa: list(dB),R2: dB] :
( ( foldl(dB,dB,app,abs(R2),Rsa) = foldl(dB,dB,app,abs(S1),Ss1) )
<=> ( ( R2 = S1 )
& ( Rsa = Ss1 ) ) ) ).
tff(fact_63_dB_Osimps_I9_J,axiom,
! [DB2: dB,DB1: dB,DB4: dB] : abs(DB4) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2) ).
tff(fact_64_dB_Osimps_I8_J,axiom,
! [DB4: dB,DB2: dB,DB1: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2) != abs(DB4) ).
tff(fact_65_dB_Osimps_I7_J,axiom,
! [Nat: nat,DB4: dB] : abs(DB4) != var(Nat) ).
tff(fact_66_dB_Osimps_I6_J,axiom,
! [DB4: dB,Nat: nat] : var(Nat) != abs(DB4) ).
tff(fact_67_dB_Osimps_I12_J,axiom,
! [A: $tType,DB3: dB,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,abs(DB3)) = aa(dB,A,F3,DB3) ).
tff(fact_68_Suc__inject,axiom,
! [Y: nat,X1: nat] :
( ( suc(X1) = suc(Y) )
=> ( X1 = Y ) ) ).
tff(fact_69_Suc__n__not__n,axiom,
! [N: nat] : suc(N) != N ).
tff(fact_70_n__not__Suc__n,axiom,
! [N: nat] : N != suc(N) ).
tff(fact_71_Abs__App__neq__Var__apps,axiom,
! [Ss1: list(dB),Na: nat,Ta: dB,S1: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,abs(S1)),Ta) != foldl(dB,dB,app,var(Na),Ss1) ).
tff(fact_72_Var__apps__neq__Abs__apps,axiom,
! [Ss1: list(dB),R2: dB,Ts: list(dB),Na: nat] : foldl(dB,dB,app,var(Na),Ts) != foldl(dB,dB,app,abs(R2),Ss1) ).
tff(fact_73_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X: A] : aa(A,B,F,X) = aa(A,B,G,X)
=> ( F = G ) ) ).
tff(fact_74_Suc__neq__Zero,axiom,
! [M: nat] : suc(M) != zero_zero(nat) ).
tff(fact_75_Zero__neq__Suc,axiom,
! [M: nat] : zero_zero(nat) != suc(M) ).
tff(fact_76_nat_Osimps_I3_J,axiom,
! [Nat3: nat] : suc(Nat3) != zero_zero(nat) ).
tff(fact_77_Suc__not__Zero,axiom,
! [M: nat] : suc(M) != zero_zero(nat) ).
tff(fact_78_nat_Osimps_I2_J,axiom,
! [Nat2: nat] : zero_zero(nat) != suc(Nat2) ).
tff(fact_79_Zero__not__Suc,axiom,
! [M: nat] : zero_zero(nat) != suc(M) ).
tff(fact_80_IT_Osimps,axiom,
! [Aa: dB] :
( pp(aa(dB,bool,it,Aa))
<=> ( ? [Rs: list(dB),N2: nat] :
( ( Aa = foldl(dB,dB,app,var(N2),Rs) )
& listsp(dB,it,Rs) )
| ? [R1: dB] :
( ( Aa = abs(R1) )
& pp(aa(dB,bool,it,R1)) )
| ? [R1: dB,S: dB,Ss: list(dB)] :
( ( Aa = foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(R1)),S),Ss) )
& pp(aa(dB,bool,it,foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,R1),S),zero_zero(nat)),Ss)))
& pp(aa(dB,bool,it,S)) ) ) ) ).
tff(fact_81_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero(nat) )
=> ~ ! [Nat1: nat] : Y != suc(Nat1) ) ).
tff(fact_82_zero__induct,axiom,
! [K: nat,P1: fun(nat,bool)] :
( pp(aa(nat,bool,P1,K))
=> ( ! [N1: nat] :
( pp(aa(nat,bool,P1,suc(N1)))
=> pp(aa(nat,bool,P1,N1)) )
=> pp(aa(nat,bool,P1,zero_zero(nat))) ) ) ).
tff(fact_83_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ? [M2: nat] : N = suc(M2) ) ).
tff(fact_84_nat__induct,axiom,
! [Na: nat,P1: fun(nat,bool)] :
( pp(aa(nat,bool,P1,zero_zero(nat)))
=> ( ! [N1: nat] :
( pp(aa(nat,bool,P1,N1))
=> pp(aa(nat,bool,P1,suc(N1))) )
=> pp(aa(nat,bool,P1,Na)) ) ) ).
tff(fact_85_dB_Osize_I3_J,axiom,
! [DB: dB] : dB_size(abs(DB)) = plus_plus(nat,dB_size(DB),suc(zero_zero(nat))) ).
tff(fact_86_dB_Osize_I2_J,axiom,
! [DB2: dB,DB1: dB] : dB_size(aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2)) = plus_plus(nat,plus_plus(nat,dB_size(DB1),dB_size(DB2)),suc(zero_zero(nat))) ).
tff(fact_87_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,Aa: A,Ba: A] :
( ( plus_plus(A,Ba,Aa) = plus_plus(A,C1,Aa) )
<=> ( Ba = C1 ) ) ) ).
tff(fact_88_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,Ba: A,Aa: A] :
( ( plus_plus(A,Aa,Ba) = plus_plus(A,Aa,C1) )
<=> ( Ba = C1 ) ) ) ).
tff(fact_89_nat__add__right__cancel,axiom,
! [Na: nat,K: nat,M1: nat] :
( ( plus_plus(nat,M1,K) = plus_plus(nat,Na,K) )
<=> ( M1 = Na ) ) ).
tff(fact_90_nat__add__left__cancel,axiom,
! [Na: nat,M1: nat,K: nat] :
( ( plus_plus(nat,K,M1) = plus_plus(nat,K,Na) )
<=> ( M1 = Na ) ) ).
tff(fact_91_double__zero__sym,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [Aa: A] :
( ( zero_zero(A) = plus_plus(A,Aa,Aa) )
<=> ( Aa = zero_zero(A) ) ) ) ).
tff(fact_92_add__is__0,axiom,
! [Na: nat,M1: nat] :
( ( plus_plus(nat,M1,Na) = zero_zero(nat) )
<=> ( ( M1 = zero_zero(nat) )
& ( Na = zero_zero(nat) ) ) ) ).
tff(fact_93_add__Suc__right,axiom,
! [N: nat,M: nat] : plus_plus(nat,M,suc(N)) = suc(plus_plus(nat,M,N)) ).
tff(fact_94_add__Suc,axiom,
! [N: nat,M: nat] : plus_plus(nat,suc(M),N) = suc(plus_plus(nat,M,N)) ).
tff(fact_95_nat_Osize_I4_J,axiom,
! [Nat: nat] : size_size(nat,suc(Nat)) = plus_plus(nat,size_size(nat,Nat),suc(zero_zero(nat))) ).
tff(fact_96_nat_Osize_I2_J,axiom,
! [Nat: nat] : nat_size(suc(Nat)) = plus_plus(nat,nat_size(Nat),suc(zero_zero(nat))) ).
tff(fact_97_dB_Osize_I5_J,axiom,
! [DB2: dB,DB1: dB] : size_size(dB,aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2)) = plus_plus(nat,plus_plus(nat,size_size(dB,DB1),size_size(dB,DB2)),suc(zero_zero(nat))) ).
%----Arities (3)
tff(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
tff(arity_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
%----Helper facts (14)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),combb(B,C,A),P),Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBI_1_1_U,axiom,
! [A: $tType,P: A] : aa(A,A,combi(A),P) = P ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,combk(A,B,P),Q) = P ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa(A,C,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(Q) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),lift,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,b),u),i)),zero_zero(nat)))) ).
%------------------------------------------------------------------------------