TPTP Problem File: LCL744_5.p
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%------------------------------------------------------------------------------
% File : LCL744_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 19
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_19 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 153 ( 52 unt; 44 typ; 0 def)
% Number of atoms : 210 ( 125 equ)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 137 ( 36 ~; 3 |; 10 &)
% ( 18 <=>; 70 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 52 ( 25 >; 27 *; 0 +; 0 <<)
% Number of predicates : 13 ( 12 usr; 0 prp; 1-4 aty)
% Number of functors : 28 ( 28 usr; 6 con; 0-5 aty)
% Number of variables : 315 ( 293 !; 1 ?; 315 :)
% ( 21 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:14:14
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
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_Type_Otype,type,
type: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (38)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
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_InductTermi_OIT,type,
it: dB > $o ).
tff(sy_c_Lambda_Obeta,type,
beta: fun(dB,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_OdB__case,type,
dB_case:
!>[T5: $tType] : ( ( fun(nat,T5) * fun(dB,fun(dB,T5)) * fun(dB,T5) * dB ) > T5 ) ).
tff(sy_c_Lambda_OdB_OdB__size,type,
dB_size: dB > nat ).
tff(sy_c_Lambda_Olift,type,
lift: ( dB * nat ) > dB ).
tff(sy_c_Lambda_Oliftn,type,
liftn: ( nat * dB * nat ) > dB ).
tff(sy_c_Lambda_Osubst,type,
subst: ( dB * dB * nat ) > dB ).
tff(sy_c_Lambda_Osubstn,type,
substn: ( dB * dB * nat ) > dB ).
tff(sy_c_ListOrder_Ostep1,type,
step1:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Ofoldl,type,
foldl:
!>[B1: $tType,A: $tType] : ( ( fun(B1,fun(A,B1)) * B1 * list(A) ) > B1 ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_Nat_OSuc,type,
suc: nat > nat ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri532925092at_aux:
!>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Type_Oshift,type,
shift:
!>[A: $tType] : ( ( fun(nat,A) * nat * A ) > fun(nat,A) ) ).
tff(sy_c_Type_Otype_OAtom,type,
atom: nat > type ).
tff(sy_c_Type_Otype_OFun,type,
fun1: ( type * type ) > type ).
tff(sy_c_Type_Otype_Otype__case,type,
type_case:
!>[T5: $tType] : ( ( fun(nat,T5) * fun(type,fun(type,T5)) * type ) > T5 ) ).
tff(sy_c_Type_Otype_Otype__size,type,
type_size: type > nat ).
tff(sy_c_Type_Otyping,type,
typing: ( fun(nat,type) * dB * type ) > $o ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B1: $tType] : ( ( fun(A,B1) * A ) > B1 ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_i,type,
i: nat ).
tff(sy_v_t,type,
t: dB ).
%----Relevant facts (98)
tff(fact_0_Lambda,axiom,
! [R2: dB] :
( it(R2)
=> it(abs(R2)) ) ).
tff(fact_1_subst__lift,axiom,
! [S1: dB,K: nat,T1: dB] : subst(lift(T1,K),S1,K) = T1 ).
tff(fact_2_lift_Osimps_I2_J,axiom,
! [K: nat,T1: dB,S1: dB] : lift(aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T1),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,lift(S1,K)),lift(T1,K)) ).
tff(fact_3_lift__preserves__beta,axiom,
! [I: nat,S1: dB,R2: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R2),S1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,lift(R2,I)),lift(S1,I))) ) ).
tff(fact_4_liftn__lift,axiom,
! [K: nat,T1: dB,N1: nat] : liftn(suc(N1),T1,K) = lift(liftn(N1,T1,K),K) ).
tff(fact_5_lift__type,axiom,
! [U4: type,Ia: nat,T2: type,Ta: dB,E: fun(nat,type)] :
( typing(E,Ta,T2)
=> typing(shift(type,E,Ia,U4),lift(Ta,Ia),T2) ) ).
tff(fact_6_dB_Osimps_I2_J,axiom,
! [DB22: dB,DB12: 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,DB12),DB22) )
<=> ( ( DB11 = DB12 )
& ( DB21 = DB22 ) ) ) ).
tff(fact_7_dB_Osimps_I3_J,axiom,
! [DB5: dB,DB3: dB] :
( ( abs(DB3) = abs(DB5) )
<=> ( DB3 = DB5 ) ) ).
tff(fact_8_appL,axiom,
! [U1: dB,T1: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),U1)),aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U1))) ) ).
tff(fact_9_appR,axiom,
! [U1: dB,T1: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,U1),S1)),aa(dB,dB,aa(dB,fun(dB,dB),app,U1),T1))) ) ).
tff(fact_10_subst__App,axiom,
! [K: nat,S1: dB,U1: dB,T1: dB] : subst(aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U1),S1,K) = aa(dB,dB,aa(dB,fun(dB,dB),app,subst(T1,S1,K)),subst(U1,S1,K)) ).
tff(fact_11_abs,axiom,
! [T1: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,abs(S1)),abs(T1))) ) ).
tff(fact_12_liftn_Osimps_I2_J,axiom,
! [K: nat,T1: dB,S1: dB,N1: nat] : liftn(N1,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T1),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,liftn(N1,S1,K)),liftn(N1,T1,K)) ).
tff(fact_13_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_14_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_15_subst__preserves__beta,axiom,
! [I: nat,T1: dB,S1: dB,R2: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R2),S1))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,subst(R2,T1,I)),subst(S1,T1,I))) ) ).
tff(fact_16_shift__eq,axiom,
! [A: $tType,T2: A,E: fun(nat,A),J: nat,Ia: nat] :
( ( Ia = J )
=> ( aa(nat,A,shift(A,E,Ia,T2),J) = T2 ) ) ).
tff(fact_17_subject__reduction,axiom,
! [T4: dB,T2: type,Ta: dB,E: fun(nat,type)] :
( typing(E,Ta,T2)
=> ( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,Ta),T4))
=> typing(E,T4,T2) ) ) ).
tff(fact_18_subst__lemma,axiom,
! [Ia: nat,U4: type,U3: dB,E1: fun(nat,type),T2: type,Ta: dB,E: fun(nat,type)] :
( typing(E,Ta,T2)
=> ( typing(E1,U3,U4)
=> ( ( E = shift(type,E1,Ia,U4) )
=> typing(E1,subst(Ta,U3,Ia),T2) ) ) ) ).
tff(fact_19_beta__cases_I2_J,axiom,
! [S1: dB,R2: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,abs(R2)),S1))
=> ~ ! [T: dB] :
( ( S1 = abs(T) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R2),T)) ) ) ).
tff(fact_20_nat_Oinject,axiom,
! [Nat3: nat,Nat2: nat] :
( ( suc(Nat2) = suc(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_21_beta,axiom,
! [T1: dB,S1: dB] : pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(S1)),T1)),subst(S1,T1,zero_zero(nat)))) ).
tff(fact_22_App,axiom,
! [Ta: dB,U4: type,T2: type,S: dB,Env: fun(nat,type)] :
( typing(Env,S,fun1(T2,U4))
=> ( typing(Env,Ta,T2)
=> typing(Env,aa(dB,dB,aa(dB,fun(dB,dB),app,S),Ta),U4) ) ) ).
tff(fact_23_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_24_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_25_substn_Osimps_I2_J,axiom,
! [K: nat,S1: dB,U1: dB,T1: dB] : substn(aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U1),S1,K) = aa(dB,dB,aa(dB,fun(dB,dB),app,substn(T1,S1,K)),substn(U1,S1,K)) ).
tff(fact_26_n__not__Suc__n,axiom,
! [N1: nat] : N1 != suc(N1) ).
tff(fact_27_Suc__n__not__n,axiom,
! [N1: nat] : suc(N1) != N1 ).
tff(fact_28_type_Osimps_I2_J,axiom,
! [Type24: type,Type14: type,Type23: type,Type13: type] :
( ( fun1(Type13,Type23) = fun1(Type14,Type24) )
<=> ( ( Type13 = Type14 )
& ( Type23 = Type24 ) ) ) ).
tff(fact_29_shift__commute,axiom,
! [A: $tType,T2: A,U4: A,Ia: nat,E: fun(nat,A)] : shift(A,shift(A,E,Ia,U4),zero_zero(nat),T2) = shift(A,shift(A,E,zero_zero(nat),T2),suc(Ia),U4) ).
tff(fact_30_substn__subst__n,axiom,
! [N1: nat,S1: dB,T1: dB] : substn(T1,S1,N1) = subst(T1,liftn(N1,S1,zero_zero(nat)),N1) ).
tff(fact_31_Abs,axiom,
! [U4: type,Ta: dB,T2: type,Env: fun(nat,type)] :
( typing(shift(type,Env,zero_zero(nat),T2),Ta,U4)
=> typing(Env,abs(Ta),fun1(T2,U4)) ) ).
tff(fact_32_Suc__neq__Zero,axiom,
! [M1: nat] : suc(M1) != zero_zero(nat) ).
tff(fact_33_Zero__neq__Suc,axiom,
! [M1: nat] : zero_zero(nat) != suc(M1) ).
tff(fact_34_nat_Osimps_I3_J,axiom,
! [Nat5: nat] : suc(Nat5) != zero_zero(nat) ).
tff(fact_35_Suc__not__Zero,axiom,
! [M1: nat] : suc(M1) != zero_zero(nat) ).
tff(fact_36_nat_Osimps_I2_J,axiom,
! [Nat4: nat] : zero_zero(nat) != suc(Nat4) ).
tff(fact_37_Zero__not__Suc,axiom,
! [M1: nat] : zero_zero(nat) != suc(M1) ).
tff(fact_38_substn__subst__0,axiom,
! [S1: dB,T1: dB] : substn(T1,S1,zero_zero(nat)) = subst(T1,S1,zero_zero(nat)) ).
tff(fact_39_liftn__0,axiom,
! [K: nat,T1: dB] : liftn(zero_zero(nat),T1,K) = T1 ).
tff(fact_40_Suc__inject,axiom,
! [Y1: nat,X1: nat] :
( ( suc(X1) = suc(Y1) )
=> ( X1 = Y1 ) ) ).
tff(fact_41_typing__elims_I3_J,axiom,
! [T2: type,Ta: dB,E: fun(nat,type)] :
( typing(E,abs(Ta),T2)
=> ~ ! [T3: type,U2: type] :
( ( T2 = fun1(T3,U2) )
=> ~ typing(shift(type,E,zero_zero(nat),T3),Ta,U2) ) ) ).
tff(fact_42_typing__elims_I2_J,axiom,
! [T2: type,U3: dB,Ta: dB,E: fun(nat,type)] :
( typing(E,aa(dB,dB,aa(dB,fun(dB,dB),app,Ta),U3),T2)
=> ~ ! [T3: type] :
( typing(E,Ta,fun1(T3,T2))
=> ~ typing(E,U3,T3) ) ) ).
tff(fact_43_abs__typeE,axiom,
! [T2: type,Ta: dB,E: fun(nat,type)] :
( typing(E,abs(Ta),T2)
=> ~ ! [U2: type,V: type] : ~ typing(shift(type,E,zero_zero(nat),U2),Ta,V) ) ).
tff(fact_44_beta__cases_I3_J,axiom,
! [U1: dB,T1: dB,S1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T1)),U1))
=> ( ! [S2: dB] :
( ( U1 = subst(S2,T1,zero_zero(nat)) )
=> ( S1 != abs(S2) ) )
=> ( ! [T: dB] :
( ( U1 = aa(dB,dB,aa(dB,fun(dB,dB),app,T),T1) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,S1),T)) )
=> ~ ! [T: dB] :
( ( U1 = aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T) )
=> ~ pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,T1),T)) ) ) ) ) ).
tff(fact_45_nat_Oexhaust,axiom,
! [Y1: nat] :
( ( Y1 != zero_zero(nat) )
=> ~ ! [Nat: nat] : Y1 != suc(Nat) ) ).
tff(fact_46_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X: A] :
( ( zero_zero(A) = X )
<=> ( X = zero_zero(A) ) ) ) ).
tff(fact_47_not0__implies__Suc,axiom,
! [N1: nat] :
( ( N1 != zero_zero(nat) )
=> ? [M2: nat] : N1 = suc(M2) ) ).
tff(fact_48_nat__induct,axiom,
! [N: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,P,zero_zero(nat)))
=> ( ! [N2: nat] :
( pp(aa(nat,bool,P,N2))
=> pp(aa(nat,bool,P,suc(N2))) )
=> pp(aa(nat,bool,P,N)) ) ) ).
tff(fact_49_zero__induct,axiom,
! [K1: nat,P: fun(nat,bool)] :
( pp(aa(nat,bool,P,K1))
=> ( ! [N2: nat] :
( pp(aa(nat,bool,P,suc(N2)))
=> pp(aa(nat,bool,P,N2)) )
=> pp(aa(nat,bool,P,zero_zero(nat))) ) ) ).
tff(fact_50_Beta,axiom,
! [Ss: list(dB),S: dB,R: dB] :
( it(foldl(dB,dB,app,subst(R,S,zero_zero(nat)),Ss))
=> ( it(S)
=> it(foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(R)),S),Ss)) ) ) ).
tff(fact_51_Abs__apps__eq__Abs__apps__conv,axiom,
! [Ss: list(dB),S: dB,Rs: list(dB),R: dB] :
( ( foldl(dB,dB,app,abs(R),Rs) = foldl(dB,dB,app,abs(S),Ss) )
<=> ( ( R = S )
& ( Rs = Ss ) ) ) ).
tff(fact_52_apps__eq__tail__conv,axiom,
! [S: dB,Ts: list(dB),R: dB] :
( ( foldl(dB,dB,app,R,Ts) = foldl(dB,dB,app,S,Ts) )
<=> ( R = S ) ) ).
tff(fact_53_apps__preserves__beta,axiom,
! [Ss: list(dB),S: dB,R: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R),S))
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,R,Ss)),foldl(dB,dB,app,S,Ss))) ) ).
tff(fact_54_type_Osimps_I6_J,axiom,
! [A: $tType,Type23: type,Type13: type,F2: fun(type,fun(type,A)),F1: fun(nat,A)] : type_case(A,F1,F2,fun1(Type13,Type23)) = aa(type,A,aa(type,fun(type,A),F2,Type13),Type23) ).
tff(fact_55_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Ia: A,N: nat,Inc: fun(A,A)] : semiri532925092at_aux(A,Inc,suc(N),Ia) = semiri532925092at_aux(A,Inc,N,aa(A,A,Inc,Ia)) ) ).
tff(fact_56_apps__preserves__betas,axiom,
! [R: dB,Ss: list(dB),Rs: list(dB)] :
( step1(dB,beta,Rs,Ss)
=> pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,R,Rs)),foldl(dB,dB,app,R,Ss))) ) ).
tff(fact_57_type_Osimps_I5_J,axiom,
! [A: $tType,Nat2: nat,F2: fun(type,fun(type,A)),F1: fun(nat,A)] : type_case(A,F1,F2,atom(Nat2)) = aa(nat,A,F1,Nat2) ).
tff(fact_58_type_Osimps_I1_J,axiom,
! [Nat3: nat,Nat2: nat] :
( ( atom(Nat2) = atom(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_59_type_Osimps_I3_J,axiom,
! [Type22: type,Type12: type,Nat1: nat] : atom(Nat1) != fun1(Type12,Type22) ).
tff(fact_60_type_Osimps_I4_J,axiom,
! [Nat1: nat,Type22: type,Type12: type] : fun1(Type12,Type22) != atom(Nat1) ).
tff(fact_61_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Ia: A,Inc: fun(A,A)] : semiri532925092at_aux(A,Inc,zero_zero(nat),Ia) = Ia ) ).
tff(fact_62_type_Osize_I1_J,axiom,
! [Nat1: nat] : type_size(atom(Nat1)) = zero_zero(nat) ).
tff(fact_63_type_Osize_I3_J,axiom,
! [Nat1: nat] : size_size(type,atom(Nat1)) = zero_zero(nat) ).
tff(fact_64_type_Oexhaust,axiom,
! [Y1: type] :
( ! [Nat: nat] : Y1 != atom(Nat)
=> ~ ! [Type11: type,Type21: type] : Y1 != fun1(Type11,Type21) ) ).
tff(fact_65_type_Osize_I4_J,axiom,
! [Type2: type,Type1: type] : size_size(type,fun1(Type1,Type2)) = plus_plus(nat,plus_plus(nat,size_size(type,Type1),size_size(type,Type2)),suc(zero_zero(nat))) ).
tff(fact_66_type_Osize_I2_J,axiom,
! [Type2: type,Type1: type] : type_size(fun1(Type1,Type2)) = plus_plus(nat,plus_plus(nat,type_size(Type1),type_size(Type2)),suc(zero_zero(nat))) ).
tff(fact_67_apps__betasE,axiom,
! [S: dB,Rs: list(dB),R: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,foldl(dB,dB,app,R,Rs)),S))
=> ( ! [R1: dB] :
( pp(aa(dB,bool,aa(dB,fun(dB,bool),beta,R),R1))
=> ( S != foldl(dB,dB,app,R1,Rs) ) )
=> ( ! [Rs1: list(dB)] :
( step1(dB,beta,Rs,Rs1)
=> ( S != foldl(dB,dB,app,R,Rs1) ) )
=> ~ ! [T: dB] :
( ( R = abs(T) )
=> ! [U: dB,Us: list(dB)] :
( ( Rs = cons(dB,U,Us) )
=> ( S != foldl(dB,dB,app,subst(T,U,zero_zero(nat)),Us) ) ) ) ) ) ) ).
tff(fact_68_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B2: A,A1: A] :
( ( plus_plus(A,A1,B2) = plus_plus(A,A1,C1) )
<=> ( B2 = C1 ) ) ) ).
tff(fact_69_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,A1: A,B2: A] :
( ( plus_plus(A,B2,A1) = plus_plus(A,C1,A1) )
<=> ( B2 = C1 ) ) ) ).
tff(fact_70_nat__add__left__cancel,axiom,
! [N: nat,M: nat,K1: nat] :
( ( plus_plus(nat,K1,M) = plus_plus(nat,K1,N) )
<=> ( M = N ) ) ).
tff(fact_71_nat__add__right__cancel,axiom,
! [N: nat,K1: nat,M: nat] :
( ( plus_plus(nat,M,K1) = plus_plus(nat,N,K1) )
<=> ( M = N ) ) ).
tff(fact_72_double__zero__sym,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ( zero_zero(A) = plus_plus(A,A1,A1) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_73_ext,axiom,
! [B1: $tType,A: $tType,G: fun(A,B1),F: fun(A,B1)] :
( ! [X2: A] : aa(A,B1,F,X2) = aa(A,B1,G,X2)
=> ( F = G ) ) ).
tff(fact_74_add__is__0,axiom,
! [N: nat,M: nat] :
( ( plus_plus(nat,M,N) = zero_zero(nat) )
<=> ( ( M = zero_zero(nat) )
& ( N = zero_zero(nat) ) ) ) ).
tff(fact_75_add__Suc__right,axiom,
! [N1: nat,M1: nat] : plus_plus(nat,M1,suc(N1)) = suc(plus_plus(nat,M1,N1)) ).
tff(fact_76_add__Suc,axiom,
! [N1: nat,M1: nat] : plus_plus(nat,suc(M1),N1) = suc(plus_plus(nat,M1,N1)) ).
tff(fact_77_add_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A2: A] : plus_plus(A,A2,zero_zero(A)) = A2 ) ).
tff(fact_78_add__0__right,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A2: A] : plus_plus(A,A2,zero_zero(A)) = A2 ) ).
tff(fact_79_add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A2: A] : plus_plus(A,zero_zero(A),A2) = A2 ) ).
tff(fact_80_add__0__left,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A2: A] : plus_plus(A,zero_zero(A),A2) = A2 ) ).
tff(fact_81_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C: A,B: A,A2: A] : plus_plus(A,plus_plus(A,A2,B),C) = plus_plus(A,A2,plus_plus(A,B,C)) ) ).
tff(fact_82_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,B: A,A2: A] :
( ( plus_plus(A,A2,B) = plus_plus(A,A2,C) )
=> ( B = C ) ) ) ).
tff(fact_83_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C: A,B: A,A2: A] :
( ( plus_plus(A,A2,B) = plus_plus(A,A2,C) )
=> ( B = C ) ) ) ).
tff(fact_84_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,A2: A,B: A] :
( ( plus_plus(A,B,A2) = plus_plus(A,C,A2) )
=> ( B = C ) ) ) ).
tff(fact_85_plus__nat_Oadd__0,axiom,
! [N1: nat] : plus_plus(nat,zero_zero(nat),N1) = N1 ).
tff(fact_86_Nat_Oadd__0__right,axiom,
! [M1: nat] : plus_plus(nat,M1,zero_zero(nat)) = M1 ).
tff(fact_87_add__eq__self__zero,axiom,
! [N1: nat,M1: nat] :
( ( plus_plus(nat,M1,N1) = M1 )
=> ( N1 = zero_zero(nat) ) ) ).
tff(fact_88_add__Suc__shift,axiom,
! [N1: nat,M1: nat] : plus_plus(nat,suc(M1),N1) = plus_plus(nat,M1,suc(N1)) ).
tff(fact_89_nat__add__commute,axiom,
! [N1: nat,M1: nat] : plus_plus(nat,M1,N1) = plus_plus(nat,N1,M1) ).
tff(fact_90_nat__add__left__commute,axiom,
! [Z: nat,Y1: nat,X1: nat] : plus_plus(nat,X1,plus_plus(nat,Y1,Z)) = plus_plus(nat,Y1,plus_plus(nat,X1,Z)) ).
tff(fact_91_nat__add__assoc,axiom,
! [K: nat,N1: nat,M1: nat] : plus_plus(nat,plus_plus(nat,M1,N1),K) = plus_plus(nat,M1,plus_plus(nat,N1,K)) ).
tff(fact_92_one__is__add,axiom,
! [N: nat,M: nat] :
( ( suc(zero_zero(nat)) = plus_plus(nat,M,N) )
<=> ( ( ( M = suc(zero_zero(nat)) )
& ( N = zero_zero(nat) ) )
| ( ( M = zero_zero(nat) )
& ( N = suc(zero_zero(nat)) ) ) ) ) ).
tff(fact_93_add__is__1,axiom,
! [N: nat,M: nat] :
( ( plus_plus(nat,M,N) = suc(zero_zero(nat)) )
<=> ( ( ( M = suc(zero_zero(nat)) )
& ( N = zero_zero(nat) ) )
| ( ( M = zero_zero(nat) )
& ( N = suc(zero_zero(nat)) ) ) ) ) ).
tff(fact_94_Cons__step1__Cons,axiom,
! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A,R: fun(A,fun(A,bool))] :
( step1(A,R,cons(A,Y,Ys),cons(A,X,Xs))
<=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),R,Y),X))
& ( Xs = Ys ) )
| ( ( X = Y )
& step1(A,R,Ys,Xs) ) ) ) ).
tff(fact_95_double__eq__0__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ( plus_plus(A,A1,A1) = zero_zero(A) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_96_dB_Osize_I3_J,axiom,
! [DB: dB] : dB_size(abs(DB)) = plus_plus(nat,dB_size(DB),suc(zero_zero(nat))) ).
tff(fact_97_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))) ).
%----Arities (7)
tff(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
tff(arity_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(nat) ).
tff(arity_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
it(t) ).
tff(conj_1,conjecture,
it(lift(t,i)) ).
%------------------------------------------------------------------------------