TPTP Problem File: LCL792_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL792_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 136
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_136 [Bla13]
% Status : Theorem
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 182 ( 63 unt; 46 typ; 0 def)
% Number of atoms : 287 ( 85 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 163 ( 12 ~; 5 |; 13 &)
% ( 32 <=>; 101 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 4 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 35 ( 21 >; 14 *; 0 +; 0 <<)
% Number of predicates : 16 ( 15 usr; 0 prp; 1-3 aty)
% Number of functors : 24 ( 24 usr; 5 con; 0-4 aty)
% Number of variables : 260 ( 230 !; 5 ?; 260 :)
% ( 25 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:19:58
%------------------------------------------------------------------------------
%----Should-be-implicit typings (8)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_Lambda_OdB,type,
dB: $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 (38)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Osgn__if,type,
sgn_if:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( 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_If,type,
if:
!>[A: $tType] : ( ( bool * A * A ) > A ) ).
tff(sy_c_InductTermi_OIT,type,
it: dB > $o ).
tff(sy_c_Int_Onat,type,
nat1: int > nat ).
tff(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
tff(sy_c_Lambda_Olift,type,
lift: ( dB * nat ) > dB ).
tff(sy_c_Nat_Onat_Onat__case,type,
nat_case:
!>[T1: $tType] : ( ( T1 * fun(nat,T1) * nat ) > T1 ) ).
tff(sy_c_Nat_Onat_Onat__size,type,
nat_size: nat > nat ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
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_Nat__Transfer_Otsub,type,
nat_tsub: ( int * int ) > int ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( ( A * A ) > $o ) ).
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_pp,type,
pp: bool > $o ).
tff(sy_v_t____,type,
t: dB ).
tff(sy_v_u____,type,
u: dB ).
tff(sy_v_ua______,type,
ua: dB ).
%----Relevant facts (99)
tff(fact_0__096IT_At_096,axiom,
it(t) ).
tff(fact_1_Var_I3_J,axiom,
it(ua) ).
tff(fact_2_uIT,axiom,
it(u) ).
tff(fact_3_lift__IT,axiom,
! [I: nat,T: dB] :
( it(T)
=> it(lift(T,I)) ) ).
tff(fact_4_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X1: A] :
( ( zero_zero(A) = X1 )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_5_bool_Osize_I1_J,axiom,
bool_size(fTrue) = zero_zero(nat) ).
tff(fact_6_bool_Osize_I2_J,axiom,
bool_size(fFalse) = zero_zero(nat) ).
tff(fact_7_char__size,axiom,
! [C1: char1] : char_size(C1) = zero_zero(nat) ).
tff(fact_8_nat_Osize_I1_J,axiom,
nat_size(zero_zero(nat)) = zero_zero(nat) ).
tff(fact_9_size__char,axiom,
! [C1: char1] : size_size(char1,C1) = zero_zero(nat) ).
tff(fact_10_bool_Osize_I3_J,axiom,
size_size(bool,fTrue) = zero_zero(nat) ).
tff(fact_11_bool_Osize_I4_J,axiom,
size_size(bool,fFalse) = zero_zero(nat) ).
tff(fact_12_size__literal__def,axiom,
! [S: literal] : size_size(literal,S) = zero_zero(nat) ).
tff(fact_13_nat_Osize_I3_J,axiom,
size_size(nat,zero_zero(nat)) = zero_zero(nat) ).
tff(fact_14_of__nat__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,zero_zero(nat)) = zero_zero(A) ) ) ).
tff(fact_15_of__nat__eq__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [Na: nat,M: nat] :
( ( semiring_1_of_nat(A,M) = semiring_1_of_nat(A,Na) )
<=> ( M = Na ) ) ) ).
tff(fact_16_nat__size,axiom,
! [N: nat] : size_size(nat,N) = N ).
tff(fact_17_size__bool,axiom,
! [B1: bool] : size_size(bool,B1) = zero_zero(nat) ).
tff(fact_18_char_Osize_I2_J,axiom,
! [Nibble23: nibble,Nibble13: nibble] : size_size(char1,char(Nibble13,Nibble23)) = zero_zero(nat) ).
tff(fact_19_char_Osize_I1_J,axiom,
! [Nibble23: nibble,Nibble13: nibble] : char_size(char(Nibble13,Nibble23)) = zero_zero(nat) ).
tff(fact_20_transfer__int__nat__numerals_I1_J,axiom,
zero_zero(int) = semiring_1_of_nat(int,zero_zero(nat)) ).
tff(fact_21_int__0,axiom,
semiring_1_of_nat(int,zero_zero(nat)) = zero_zero(int) ).
tff(fact_22_int__eq__0__conv,axiom,
! [Na: nat] :
( ( semiring_1_of_nat(int,Na) = zero_zero(int) )
<=> ( Na = zero_zero(nat) ) ) ).
tff(fact_23_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Ib: A,Inc: fun(A,A)] : semiri532925092at_aux(A,Inc,zero_zero(nat),Ib) = Ib ) ).
tff(fact_24_nat__case__0,axiom,
! [A: $tType,F2: fun(nat,A),F1: A] : nat_case(A,F1,F2,zero_zero(nat)) = F1 ).
tff(fact_25_sgn0,axiom,
! [A: $tType] :
( sgn_if(A)
=> ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_26_char_Oinject,axiom,
! [Nibble22: nibble,Nibble12: nibble,Nibble2: nibble,Nibble1: nibble] :
( ( char(Nibble1,Nibble2) = char(Nibble12,Nibble22) )
<=> ( ( Nibble1 = Nibble12 )
& ( Nibble2 = Nibble22 ) ) ) ).
tff(fact_27_int__int__eq,axiom,
! [Na: nat,M: nat] :
( ( semiring_1_of_nat(int,M) = semiring_1_of_nat(int,Na) )
<=> ( M = Na ) ) ).
tff(fact_28_transfer__int__nat__relations_I1_J,axiom,
! [Y1: nat,X1: nat] :
( ( semiring_1_of_nat(int,X1) = semiring_1_of_nat(int,Y1) )
<=> ( X1 = Y1 ) ) ).
tff(fact_29_int__if__cong,axiom,
! [Y1: nat,X1: nat,P1: bool] :
( ( pp(P1)
=> ( semiring_1_of_nat(int,X1) = semiring_1_of_nat(int,if(nat,P1,X1,Y1)) ) )
& ( ~ pp(P1)
=> ( semiring_1_of_nat(int,Y1) = semiring_1_of_nat(int,if(nat,P1,X1,Y1)) ) ) ) ).
tff(fact_30_sgn__sgn,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : sgn_sgn(A,sgn_sgn(A,A1)) = sgn_sgn(A,A1) ) ).
tff(fact_31_sgn__0__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Aa: A] :
( ( sgn_sgn(A,Aa) = zero_zero(A) )
<=> ( Aa = zero_zero(A) ) ) ) ).
tff(fact_32_char_Oexhaust,axiom,
! [Y: char1] :
~ ! [Nibble11: nibble,Nibble21: nibble] : Y != char(Nibble11,Nibble21) ).
tff(fact_33_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_34_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_35_int__le__0__conv,axiom,
! [Na: nat] :
( ord_less_eq(int,semiring_1_of_nat(int,Na),zero_zero(int))
<=> ( Na = zero_zero(nat) ) ) ).
tff(fact_36_of__nat__le__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Na: nat,M: nat] :
( ord_less_eq(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,Na))
<=> ord_less_eq(nat,M,Na) ) ) ).
tff(fact_37_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq(int,zero_zero(int),zero_zero(int)) ).
tff(fact_38_zero__le__imp__of__nat,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M1: nat] : ord_less_eq(A,zero_zero(A),semiring_1_of_nat(A,M1)) ) ).
tff(fact_39_of__nat__0__le__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat] : ord_less_eq(A,zero_zero(A),semiring_1_of_nat(A,N)) ) ).
tff(fact_40_transfer__int__nat__quantifiers_I1_J,axiom,
! [P1: fun(int,bool)] :
( ! [X2: int] :
( ord_less_eq(int,zero_zero(int),X2)
=> pp(aa(int,bool,P1,X2)) )
<=> ! [X2: nat] : pp(aa(int,bool,P1,semiring_1_of_nat(int,X2))) ) ).
tff(fact_41_transfer__int__nat__quantifiers_I2_J,axiom,
! [P1: fun(int,bool)] :
( ? [X2: int] :
( ord_less_eq(int,zero_zero(int),X2)
& pp(aa(int,bool,P1,X2)) )
<=> ? [X2: nat] : pp(aa(int,bool,P1,semiring_1_of_nat(int,X2))) ) ).
tff(fact_42_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z2: nat] : ord_less_eq(int,zero_zero(int),semiring_1_of_nat(int,Z2)) ).
tff(fact_43_zero__zle__int,axiom,
! [N: nat] : ord_less_eq(int,zero_zero(int),semiring_1_of_nat(int,N)) ).
tff(fact_44_nonneg__eq__int,axiom,
! [Z2: int] :
( ord_less_eq(int,zero_zero(int),Z2)
=> ~ ! [M2: nat] : Z2 != semiring_1_of_nat(int,M2) ) ).
tff(fact_45_zero__le__imp__eq__int,axiom,
! [K: int] :
( ord_less_eq(int,zero_zero(int),K)
=> ? [N1: nat] : K = semiring_1_of_nat(int,N1) ) ).
tff(fact_46_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ord_less_eq(A,X,X) ) ).
tff(fact_47_conj__le__cong,axiom,
! [P2: bool,P1: bool,X1: int] :
( ( ord_less_eq(int,zero_zero(int),X1)
=> ( pp(P1)
<=> pp(P2) ) )
=> ( ( ord_less_eq(int,zero_zero(int),X1)
& pp(P1) )
<=> ( ord_less_eq(int,zero_zero(int),X1)
& pp(P2) ) ) ) ).
tff(fact_48_imp__le__cong,axiom,
! [P2: bool,P1: bool,X1: int] :
( ( ord_less_eq(int,zero_zero(int),X1)
=> ( pp(P1)
<=> pp(P2) ) )
=> ( ( ord_less_eq(int,zero_zero(int),X1)
=> pp(P1) )
<=> ( ord_less_eq(int,zero_zero(int),X1)
=> pp(P2) ) ) ) ).
tff(fact_49_le0,axiom,
! [N: nat] : ord_less_eq(nat,zero_zero(nat),N) ).
tff(fact_50_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ord_less_eq(nat,zero_zero(nat),N) ).
tff(fact_51_le__0__eq,axiom,
! [Na: nat] :
( ord_less_eq(nat,Na,zero_zero(nat))
<=> ( Na = zero_zero(nat) ) ) ).
tff(fact_52_le__antisym,axiom,
! [N: nat,M1: nat] :
( ord_less_eq(nat,M1,N)
=> ( ord_less_eq(nat,N,M1)
=> ( M1 = N ) ) ) ).
tff(fact_53_le__trans,axiom,
! [K: nat,J: nat,I: nat] :
( ord_less_eq(nat,I,J)
=> ( ord_less_eq(nat,J,K)
=> ord_less_eq(nat,I,K) ) ) ).
tff(fact_54_eq__imp__le,axiom,
! [N: nat,M1: nat] :
( ( M1 = N )
=> ord_less_eq(nat,M1,N) ) ).
tff(fact_55_nat__le__linear,axiom,
! [N: nat,M1: nat] :
( ord_less_eq(nat,M1,N)
| ord_less_eq(nat,N,M1) ) ).
tff(fact_56_le__refl,axiom,
! [N: nat] : ord_less_eq(nat,N,N) ).
tff(fact_57_transfer__int__nat__relations_I3_J,axiom,
! [Y1: nat,X1: nat] :
( ord_less_eq(int,semiring_1_of_nat(int,X1),semiring_1_of_nat(int,Y1))
<=> ord_less_eq(nat,X1,Y1) ) ).
tff(fact_58_zle__int,axiom,
! [Na: nat,M: nat] :
( ord_less_eq(int,semiring_1_of_nat(int,M),semiring_1_of_nat(int,Na))
<=> ord_less_eq(nat,M,Na) ) ).
tff(fact_59_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ ord_less_eq(A,X,Y)
=> ord_less_eq(A,Y,X) ) ) ).
tff(fact_60_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X1: A,G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
=> ord_less_eq(B,aa(A,B,F,X1),aa(A,B,G,X1)) ) ) ).
tff(fact_61_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z2: A,Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ( ord_less_eq(A,Y,Z2)
=> ord_less_eq(A,X,Z2) ) ) ) ).
tff(fact_62_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ( ord_less_eq(A,Y,X)
=> ( X = Y ) ) ) ) ).
tff(fact_63_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B2: A,A1: A] :
( ord_less_eq(A,A1,B2)
=> ( ( B2 = C1 )
=> ord_less_eq(A,A1,C1) ) ) ) ).
tff(fact_64_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B2: A,A1: A] :
( ( A1 = B2 )
=> ( ord_less_eq(A,B2,C1)
=> ord_less_eq(A,A1,C1) ) ) ) ).
tff(fact_65_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [X1: A,Y1: A] :
( ord_less_eq(A,Y1,X1)
=> ( ord_less_eq(A,X1,Y1)
<=> ( X1 = Y1 ) ) ) ) ).
tff(fact_66_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X1: A,G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
=> ord_less_eq(B,aa(A,B,F,X1),aa(A,B,G,X1)) ) ) ).
tff(fact_67_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ( X = Y )
=> ord_less_eq(A,X,Y) ) ) ).
tff(fact_68_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ( X1 = Y1 )
<=> ( ord_less_eq(A,X1,Y1)
& ord_less_eq(A,Y1,X1) ) ) ) ).
tff(fact_69_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
| ord_less_eq(A,Y,X) ) ) ).
tff(fact_70_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
<=> ! [X2: A] : ord_less_eq(B,aa(A,B,F,X2),aa(A,B,G,X2)) ) ) ).
tff(fact_71_of__int__le__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( ord_less_eq(A,ring_1_of_int(A,Z),zero_zero(A))
<=> ord_less_eq(int,Z,zero_zero(int)) ) ) ).
tff(fact_72_of__int__0__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( ord_less_eq(A,zero_zero(A),ring_1_of_int(A,Z))
<=> ord_less_eq(int,zero_zero(int),Z) ) ) ).
tff(fact_73_nonneg__int__cases,axiom,
! [K: int] :
( ord_less_eq(int,zero_zero(int),K)
=> ~ ! [N1: nat] : K != semiring_1_of_nat(int,N1) ) ).
tff(fact_74_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
! [Y: int,X: int] :
( ord_less_eq(int,zero_zero(int),X)
=> ( ord_less_eq(int,zero_zero(int),Y)
=> ord_less_eq(int,zero_zero(int),nat_tsub(X,Y)) ) ) ).
tff(fact_75_of__int__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int,W: int] :
( ( ring_1_of_int(A,W) = ring_1_of_int(A,Z) )
<=> ( W = Z ) ) ) ).
tff(fact_76_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int] :
( ( ring_1_of_int(A,Z) = zero_zero(A) )
<=> ( Z = zero_zero(int) ) ) ) ).
tff(fact_77_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int] :
( ( zero_zero(A) = ring_1_of_int(A,Z) )
<=> ( Z = zero_zero(int) ) ) ) ).
tff(fact_78_of__int__0,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,zero_zero(int)) = zero_zero(A) ) ) ).
tff(fact_79_of__int__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int,W: int] :
( ord_less_eq(A,ring_1_of_int(A,W),ring_1_of_int(A,Z))
<=> ord_less_eq(int,W,Z) ) ) ).
tff(fact_80_of__int__of__nat__eq,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat] : ring_1_of_int(A,semiring_1_of_nat(int,N)) = semiring_1_of_nat(A,N) ) ).
tff(fact_81_of__int__int__eq,axiom,
! [N: nat] : ring_1_of_int(int,semiring_1_of_nat(int,N)) = semiring_1_of_nat(int,N) ).
tff(fact_82_le__funI,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( ! [X3: A] : ord_less_eq(B,aa(A,B,F,X3),aa(A,B,G,X3))
=> ord_less_eq(fun(A,B),F,G) ) ) ).
tff(fact_83_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C: B,F: fun(A,B),B1: A,Aa: A] :
( ord_less_eq(A,Aa,B1)
=> ( ( aa(A,B,F,B1) = C )
=> ( ! [X3: A,Y2: A] :
( ord_less_eq(A,X3,Y2)
=> ord_less_eq(B,aa(A,B,F,X3),aa(A,B,F,Y2)) )
=> ord_less_eq(B,aa(A,B,F,Aa),C) ) ) ) ) ).
tff(fact_84_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,B1: B,F: fun(B,A),Aa: A] :
( ord_less_eq(A,Aa,aa(B,A,F,B1))
=> ( ord_less_eq(B,B1,C)
=> ( ! [X3: B,Y2: B] :
( ord_less_eq(B,X3,Y2)
=> ord_less_eq(A,aa(B,A,F,X3),aa(B,A,F,Y2)) )
=> ord_less_eq(A,Aa,aa(B,A,F,C)) ) ) ) ) ).
tff(fact_85_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C: B,B1: B,F: fun(B,A),Aa: A] :
( ( Aa = aa(B,A,F,B1) )
=> ( ord_less_eq(B,B1,C)
=> ( ! [X3: B,Y2: B] :
( ord_less_eq(B,X3,Y2)
=> ord_less_eq(A,aa(B,A,F,X3),aa(B,A,F,Y2)) )
=> ord_less_eq(A,Aa,aa(B,A,F,C)) ) ) ) ) ).
tff(fact_86_order__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,F: fun(A,B),B1: A,Aa: A] :
( ord_less_eq(A,Aa,B1)
=> ( ord_less_eq(B,aa(A,B,F,B1),C)
=> ( ! [X3: A,Y2: A] :
( ord_less_eq(A,X3,Y2)
=> ord_less_eq(B,aa(A,B,F,X3),aa(A,B,F,Y2)) )
=> ord_less_eq(B,aa(A,B,F,Aa),C) ) ) ) ) ).
tff(fact_87_of__nat__nat,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z2: int] :
( ord_less_eq(int,zero_zero(int),Z2)
=> ( semiring_1_of_nat(A,nat1(Z2)) = ring_1_of_int(A,Z2) ) ) ) ).
tff(fact_88_nat__0,axiom,
nat1(zero_zero(int)) = zero_zero(nat) ).
tff(fact_89_nat__0__iff,axiom,
! [Ib: int] :
( ( nat1(Ib) = zero_zero(nat) )
<=> ord_less_eq(int,Ib,zero_zero(int)) ) ).
tff(fact_90_int__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_eq(int,zero_zero(int),Z2)
=> ( semiring_1_of_nat(int,nat1(Z2)) = Z2 ) )
& ( ~ ord_less_eq(int,zero_zero(int),Z2)
=> ( semiring_1_of_nat(int,nat1(Z2)) = zero_zero(int) ) ) ) ).
tff(fact_91_transfer__nat__int__numerals_I1_J,axiom,
zero_zero(nat) = nat1(zero_zero(int)) ).
tff(fact_92_ex__nat,axiom,
! [P1: fun(nat,bool)] :
( ? [X11: nat] : pp(aa(nat,bool,P1,X11))
<=> ? [X2: int] :
( ord_less_eq(int,zero_zero(int),X2)
& pp(aa(nat,bool,P1,nat1(X2))) ) ) ).
tff(fact_93_all__nat,axiom,
! [P1: fun(nat,bool)] :
( ! [X11: nat] : pp(aa(nat,bool,P1,X11))
<=> ! [X2: int] :
( ord_less_eq(int,zero_zero(int),X2)
=> pp(aa(nat,bool,P1,nat1(X2))) ) ) ).
tff(fact_94_transfer__nat__int__relations_I1_J,axiom,
! [Y1: int,X1: int] :
( ord_less_eq(int,zero_zero(int),X1)
=> ( ord_less_eq(int,zero_zero(int),Y1)
=> ( ( nat1(X1) = nat1(Y1) )
<=> ( X1 = Y1 ) ) ) ) ).
tff(fact_95_eq__nat__nat__iff,axiom,
! [Z1: int,Z: int] :
( ord_less_eq(int,zero_zero(int),Z)
=> ( ord_less_eq(int,zero_zero(int),Z1)
=> ( ( nat1(Z) = nat1(Z1) )
<=> ( Z = Z1 ) ) ) ) ).
tff(fact_96_nat__mono,axiom,
! [Y: int,X: int] :
( ord_less_eq(int,X,Y)
=> ord_less_eq(nat,nat1(X),nat1(Y)) ) ).
tff(fact_97_nat__if__cong,axiom,
! [Y1: int,X1: int,P1: bool] :
( ( pp(P1)
=> ( nat1(X1) = nat1(if(int,P1,X1,Y1)) ) )
& ( ~ pp(P1)
=> ( nat1(Y1) = nat1(if(int,P1,X1,Y1)) ) ) ) ).
tff(fact_98_nat__int,axiom,
! [N: nat] : nat1(semiring_1_of_nat(int,N)) = N ).
%----Arities (27)
tff(arity_fun___Orderings_Opreorder,axiom,
! [T_1: $tType,T_2: $tType] :
( preorder(T_2)
=> preorder(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oord,axiom,
! [T_1: $tType,T_2: $tType] :
( ord(T_2)
=> ord(fun(T_1,T_2)) ) ).
tff(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
tff(arity_Int_Oint___Orderings_Opreorder,axiom,
preorder(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Orderings_Oorder,axiom,
order(int) ).
tff(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
tff(arity_Int_Oint___Orderings_Oord,axiom,
ord(int) ).
tff(arity_Int_Oint___Groups_Osgn__if,axiom,
sgn_if(int) ).
tff(arity_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
tff(arity_Nat_Onat___Nat_Osemiring__char__0,axiom,
semiring_char_0(nat) ).
tff(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
tff(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
%----Helper facts (9)
tff(help_If_1_1_T,axiom,
! [A: $tType,Y: A,X: A] : if(A,fTrue,X,Y) = X ).
tff(help_If_2_1_T,axiom,
! [A: $tType,Y: A,X: A] : if(A,fFalse,X,Y) = Y ).
tff(help_If_3_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
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,
it(lift(u,zero_zero(nat))) ).
%------------------------------------------------------------------------------