TPTP Problem File: NUM923+1.p
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%------------------------------------------------------------------------------
% File : NUM923+1 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 23, 100 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_100_fofmg_l23 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 79 ( 44 unt; 0 def)
% Number of atoms : 123 ( 72 equ)
% Maximal formula atoms : 3 ( 1 avg)
% Number of connectives : 54 ( 10 ~; 2 |; 2 &)
% ( 17 <=>; 23 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 5 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 3 ( 2 usr; 0 prp; 1-2 aty)
% Number of functors : 13 ( 13 usr; 4 con; 0-3 aty)
% Number of variables : 245 ( 240 !; 5 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 15:19:57
% : Encoded with monomorphized guards.
%------------------------------------------------------------------------------
%----Explicit typings (4)
fof(gsy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,axiom,
! [B_1_1,B_2_1] : is_bool(ord_less_eq_int(B_1_1,B_2_1)) ).
fof(gsy_c_Product__Type_Ocurry_000tc__Int__Oint_000tc__Int__Oint_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1,B_3_1] : is_bool(produc262399358t_bool(B_1_1,B_2_1,B_3_1)) ).
fof(gsy_c_TwoSquares__Mirabelle__yteckqlrrb_Ois__sum2sq,axiom,
! [B_1_1] : is_bool(twoSqu526106917sum2sq(B_1_1)) ).
fof(gsy_c_hAPP_000tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J_000tc__HOL__Obool,axiom,
! [B_1_1,B_2_1] : is_bool(hAPP_P603027463t_bool(B_1_1,B_2_1)) ).
%----Relevant facts (74)
fof(fact_0_xzgcda__linear__aux1,axiom,
! [A_28,R,B_25,M_1,C_21,D_6,N] : plus_plus_int(times_times_int(minus_minus_int(A_28,times_times_int(R,B_25)),M_1),times_times_int(minus_minus_int(C_21,times_times_int(R,D_6)),N)) = minus_minus_int(plus_plus_int(times_times_int(A_28,M_1),times_times_int(C_21,N)),times_times_int(R,plus_plus_int(times_times_int(B_25,M_1),times_times_int(D_6,N)))) ).
fof(fact_1_mult__diff__mult,axiom,
! [X_6,Y_6,A_29,B_26] : minus_minus_int(times_times_int(X_6,Y_6),times_times_int(A_29,B_26)) = plus_plus_int(times_times_int(X_6,minus_minus_int(Y_6,B_26)),times_times_int(minus_minus_int(X_6,A_29),B_26)) ).
fof(fact_2_eq__add__iff2,axiom,
! [Aa,E,C,Ba,D_1] :
( plus_plus_int(times_times_int(Aa,E),C) = plus_plus_int(times_times_int(Ba,E),D_1)
<=> C = plus_plus_int(times_times_int(minus_minus_int(Ba,Aa),E),D_1) ) ).
fof(fact_3_eq__add__iff1,axiom,
! [Aa,E,C,Ba,D_1] :
( plus_plus_int(times_times_int(Aa,E),C) = plus_plus_int(times_times_int(Ba,E),D_1)
<=> plus_plus_int(times_times_int(minus_minus_int(Aa,Ba),E),C) = D_1 ) ).
fof(fact_4_is__sum2sq__def,axiom,
! [X_4] :
( hBOOL(twoSqu526106917sum2sq(X_4))
<=> ? [A_5,B_5] : twoSqu536811803sum2sq(product_Pair_int_int(A_5,B_5)) = X_4 ) ).
fof(fact_5_Int2_Oaux1,axiom,
! [A_28,B_25,C_21] :
( minus_minus_int(A_28,B_25) = C_21
=> A_28 = plus_plus_int(C_21,B_25) ) ).
fof(fact_6_zdiff__zmult__distrib2,axiom,
! [W,Z1,Z2] : times_times_int(W,minus_minus_int(Z1,Z2)) = minus_minus_int(times_times_int(W,Z1),times_times_int(W,Z2)) ).
fof(fact_7_zdiff__zmult__distrib,axiom,
! [Z1,Z2,W] : times_times_int(minus_minus_int(Z1,Z2),W) = minus_minus_int(times_times_int(Z1,W),times_times_int(Z2,W)) ).
fof(fact_8_zadd__zmult__distrib2,axiom,
! [W,Z1,Z2] : times_times_int(W,plus_plus_int(Z1,Z2)) = plus_plus_int(times_times_int(W,Z1),times_times_int(W,Z2)) ).
fof(fact_9_zadd__zmult__distrib,axiom,
! [Z1,Z2,W] : times_times_int(plus_plus_int(Z1,Z2),W) = plus_plus_int(times_times_int(Z1,W),times_times_int(Z2,W)) ).
fof(fact_10_diff__add__cancel,axiom,
! [A_27,B_24] : plus_plus_int(minus_minus_int(A_27,B_24),B_24) = A_27 ).
fof(fact_11_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_26,B_23,C_20] : times_times_int(times_times_int(A_26,B_23),C_20) = times_times_int(A_26,times_times_int(B_23,C_20)) ).
fof(fact_12_add__right__imp__eq,axiom,
! [B_22,A_25,C_19] :
( plus_plus_int(B_22,A_25) = plus_plus_int(C_19,A_25)
=> B_22 = C_19 ) ).
fof(fact_13_add__imp__eq,axiom,
! [A_24,B_21,C_18] :
( plus_plus_int(A_24,B_21) = plus_plus_int(A_24,C_18)
=> B_21 = C_18 ) ).
fof(fact_14_add__left__imp__eq,axiom,
! [A_23,B_20,C_17] :
( plus_plus_int(A_23,B_20) = plus_plus_int(A_23,C_17)
=> B_20 = C_17 ) ).
fof(fact_15_add__right__cancel,axiom,
! [Ba,Aa,C] :
( plus_plus_int(Ba,Aa) = plus_plus_int(C,Aa)
<=> Ba = C ) ).
fof(fact_16_add__left__cancel,axiom,
! [Aa,Ba,C] :
( plus_plus_int(Aa,Ba) = plus_plus_int(Aa,C)
<=> Ba = C ) ).
fof(fact_17_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_22,B_19,C_16] : plus_plus_int(plus_plus_int(A_22,B_19),C_16) = plus_plus_int(A_22,plus_plus_int(B_19,C_16)) ).
fof(fact_18_diff__eq__diff__eq,axiom,
! [Aa,Ba,C,D_1] :
( minus_minus_int(Aa,Ba) = minus_minus_int(C,D_1)
=> ( Aa = Ba
<=> C = D_1 ) ) ).
fof(fact_19_zmult__assoc,axiom,
! [Z1,Z2,Z3] : times_times_int(times_times_int(Z1,Z2),Z3) = times_times_int(Z1,times_times_int(Z2,Z3)) ).
fof(fact_20_zmult__commute,axiom,
! [Z,W] : times_times_int(Z,W) = times_times_int(W,Z) ).
fof(fact_21_zadd__assoc,axiom,
! [Z1,Z2,Z3] : plus_plus_int(plus_plus_int(Z1,Z2),Z3) = plus_plus_int(Z1,plus_plus_int(Z2,Z3)) ).
fof(fact_22_zadd__left__commute,axiom,
! [X_5,Y_5,Z] : plus_plus_int(X_5,plus_plus_int(Y_5,Z)) = plus_plus_int(Y_5,plus_plus_int(X_5,Z)) ).
fof(fact_23_zadd__commute,axiom,
! [Z,W] : plus_plus_int(Z,W) = plus_plus_int(W,Z) ).
fof(fact_24_combine__common__factor,axiom,
! [A_21,E_1,B_18,C_15] : plus_plus_int(times_times_int(A_21,E_1),plus_plus_int(times_times_int(B_18,E_1),C_15)) = plus_plus_int(times_times_int(plus_plus_int(A_21,B_18),E_1),C_15) ).
fof(fact_25_comm__semiring__class_Odistrib,axiom,
! [A_20,B_17,C_14] : times_times_int(plus_plus_int(A_20,B_17),C_14) = plus_plus_int(times_times_int(A_20,C_14),times_times_int(B_17,C_14)) ).
fof(fact_26_add__diff__add,axiom,
! [A_19,C_13,B_16,D_5] : minus_minus_int(plus_plus_int(A_19,C_13),plus_plus_int(B_16,D_5)) = plus_plus_int(minus_minus_int(A_19,B_16),minus_minus_int(C_13,D_5)) ).
fof(fact_27_add__diff__cancel,axiom,
! [A_18,B_15] : minus_minus_int(plus_plus_int(A_18,B_15),B_15) = A_18 ).
fof(fact_28_crossproduct__eq,axiom,
! [W_1,Y_4,X_4,Z_2] :
( plus_plus_int(times_times_int(W_1,Y_4),times_times_int(X_4,Z_2)) = plus_plus_int(times_times_int(W_1,Z_2),times_times_int(X_4,Y_4))
<=> ( W_1 = X_4
| Y_4 = Z_2 ) ) ).
fof(fact_29_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_17,M,B_14] : plus_plus_int(times_times_int(A_17,M),times_times_int(B_14,M)) = times_times_int(plus_plus_int(A_17,B_14),M) ).
fof(fact_30_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_16,B_13,C_12] : times_times_int(plus_plus_int(A_16,B_13),C_12) = plus_plus_int(times_times_int(A_16,C_12),times_times_int(B_13,C_12)) ).
fof(fact_31_crossproduct__noteq,axiom,
! [C,D_1,Aa,Ba] :
( ( Aa != Ba
& C != D_1 )
<=> plus_plus_int(times_times_int(Aa,C),times_times_int(Ba,D_1)) != plus_plus_int(times_times_int(Aa,D_1),times_times_int(Ba,C)) ) ).
fof(fact_32_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_3,Y_3,Z_1] : times_times_int(X_3,plus_plus_int(Y_3,Z_1)) = plus_plus_int(times_times_int(X_3,Y_3),times_times_int(X_3,Z_1)) ).
fof(fact_33_Pair__inject,axiom,
! [A_15,B_12,A_14,B_11] :
( product_Pair_int_int(A_15,B_12) = product_Pair_int_int(A_14,B_11)
=> ~ ( A_15 = A_14
=> B_12 != B_11 ) ) ).
fof(fact_34_Pair__eq,axiom,
! [Aa,Ba,A_13,B_10] :
( product_Pair_int_int(Aa,Ba) = product_Pair_int_int(A_13,B_10)
<=> ( Aa = A_13
& Ba = B_10 ) ) ).
fof(fact_35_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_12,B_9] : times_times_int(A_12,B_9) = times_times_int(B_9,A_12) ).
fof(fact_36_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6,Rx_6,Ry_4] : times_times_int(Lx_6,times_times_int(Rx_6,Ry_4)) = times_times_int(Rx_6,times_times_int(Lx_6,Ry_4)) ).
fof(fact_37_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5,Rx_5,Ry_3] : times_times_int(Lx_5,times_times_int(Rx_5,Ry_3)) = times_times_int(times_times_int(Lx_5,Rx_5),Ry_3) ).
fof(fact_38_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4,Ly_4,Rx_4] : times_times_int(times_times_int(Lx_4,Ly_4),Rx_4) = times_times_int(Lx_4,times_times_int(Ly_4,Rx_4)) ).
fof(fact_39_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3,Ly_3,Rx_3] : times_times_int(times_times_int(Lx_3,Ly_3),Rx_3) = times_times_int(times_times_int(Lx_3,Rx_3),Ly_3) ).
fof(fact_40_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2,Ly_2,Rx_2,Ry_2] : times_times_int(times_times_int(Lx_2,Ly_2),times_times_int(Rx_2,Ry_2)) = times_times_int(Lx_2,times_times_int(Ly_2,times_times_int(Rx_2,Ry_2))) ).
fof(fact_41_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1,Ly_1,Rx_1,Ry_1] : times_times_int(times_times_int(Lx_1,Ly_1),times_times_int(Rx_1,Ry_1)) = times_times_int(Rx_1,times_times_int(times_times_int(Lx_1,Ly_1),Ry_1)) ).
fof(fact_42_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx,Ly,Rx,Ry] : times_times_int(times_times_int(Lx,Ly),times_times_int(Rx,Ry)) = times_times_int(times_times_int(Lx,Rx),times_times_int(Ly,Ry)) ).
fof(fact_43_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_11,C_11] : plus_plus_int(A_11,C_11) = plus_plus_int(C_11,A_11) ).
fof(fact_44_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_10,C_10,D_4] : plus_plus_int(A_10,plus_plus_int(C_10,D_4)) = plus_plus_int(C_10,plus_plus_int(A_10,D_4)) ).
fof(fact_45_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_9,C_9,D_3] : plus_plus_int(A_9,plus_plus_int(C_9,D_3)) = plus_plus_int(plus_plus_int(A_9,C_9),D_3) ).
fof(fact_46_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_8,B_8,C_8] : plus_plus_int(plus_plus_int(A_8,B_8),C_8) = plus_plus_int(A_8,plus_plus_int(B_8,C_8)) ).
fof(fact_47_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_7,B_7,C_7] : plus_plus_int(plus_plus_int(A_7,B_7),C_7) = plus_plus_int(plus_plus_int(A_7,C_7),B_7) ).
fof(fact_48_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_6,B_6,C_6,D_2] : plus_plus_int(plus_plus_int(A_6,B_6),plus_plus_int(C_6,D_2)) = plus_plus_int(plus_plus_int(A_6,C_6),plus_plus_int(B_6,D_2)) ).
fof(fact_49_split__paired__All,axiom,
! [P_1] :
( ! [X1] : hBOOL(hAPP_P603027463t_bool(P_1,X1))
<=> ! [A_5,B_5] : hBOOL(hAPP_P603027463t_bool(P_1,product_Pair_int_int(A_5,B_5))) ) ).
fof(fact_50_split__paired__Ex,axiom,
! [P_1] :
( ? [X1] : hBOOL(hAPP_P603027463t_bool(P_1,X1))
<=> ? [A_5,B_5] : hBOOL(hAPP_P603027463t_bool(P_1,product_Pair_int_int(A_5,B_5))) ) ).
fof(fact_51_prod_Oexhaust,axiom,
! [Y_2] :
~ ! [A_5,B_5] : Y_2 != product_Pair_int_int(A_5,B_5) ).
fof(fact_52_PairE,axiom,
! [P] :
~ ! [X_2,Y_1] : P != product_Pair_int_int(X_2,Y_1) ).
fof(fact_53_curryI,axiom,
! [F,Aa,Ba] :
( hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba)))
=> hBOOL(produc262399358t_bool(F,Aa,Ba)) ) ).
fof(fact_54_curryD,axiom,
! [F,Aa,Ba] :
( hBOOL(produc262399358t_bool(F,Aa,Ba))
=> hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba))) ) ).
fof(fact_55_curryE,axiom,
! [F,Aa,Ba] :
( hBOOL(produc262399358t_bool(F,Aa,Ba))
=> hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba))) ) ).
fof(fact_56_curry__conv,axiom,
! [F,Aa,Ba] :
( hBOOL(produc262399358t_bool(F,Aa,Ba))
<=> hBOOL(hAPP_P603027463t_bool(F,product_Pair_int_int(Aa,Ba))) ) ).
fof(fact_57_le__add__iff1,axiom,
! [Aa,E,C,Ba,D_1] :
( hBOOL(ord_less_eq_int(plus_plus_int(times_times_int(Aa,E),C),plus_plus_int(times_times_int(Ba,E),D_1)))
<=> hBOOL(ord_less_eq_int(plus_plus_int(times_times_int(minus_minus_int(Aa,Ba),E),C),D_1)) ) ).
fof(fact_58_le__add__iff2,axiom,
! [Aa,E,C,Ba,D_1] :
( hBOOL(ord_less_eq_int(plus_plus_int(times_times_int(Aa,E),C),plus_plus_int(times_times_int(Ba,E),D_1)))
<=> hBOOL(ord_less_eq_int(C,plus_plus_int(times_times_int(minus_minus_int(Ba,Aa),E),D_1))) ) ).
fof(fact_59_zadd__left__mono,axiom,
! [K,I,J] :
( hBOOL(ord_less_eq_int(I,J))
=> hBOOL(ord_less_eq_int(plus_plus_int(K,I),plus_plus_int(K,J))) ) ).
fof(fact_60_diff__eq__diff__less__eq,axiom,
! [Aa,Ba,C,D_1] :
( minus_minus_int(Aa,Ba) = minus_minus_int(C,D_1)
=> ( hBOOL(ord_less_eq_int(Aa,Ba))
<=> hBOOL(ord_less_eq_int(C,D_1)) ) ) ).
fof(fact_61_add__le__imp__le__left,axiom,
! [C_5,A_4,B_4] :
( hBOOL(ord_less_eq_int(plus_plus_int(C_5,A_4),plus_plus_int(C_5,B_4)))
=> hBOOL(ord_less_eq_int(A_4,B_4)) ) ).
fof(fact_62_add__le__imp__le__right,axiom,
! [A_3,C_4,B_3] :
( hBOOL(ord_less_eq_int(plus_plus_int(A_3,C_4),plus_plus_int(B_3,C_4)))
=> hBOOL(ord_less_eq_int(A_3,B_3)) ) ).
fof(fact_63_add__mono,axiom,
! [C_3,D,A_2,B_2] :
( hBOOL(ord_less_eq_int(A_2,B_2))
=> ( hBOOL(ord_less_eq_int(C_3,D))
=> hBOOL(ord_less_eq_int(plus_plus_int(A_2,C_3),plus_plus_int(B_2,D))) ) ) ).
fof(fact_64_add__left__mono,axiom,
! [C_2,A_1,B_1] :
( hBOOL(ord_less_eq_int(A_1,B_1))
=> hBOOL(ord_less_eq_int(plus_plus_int(C_2,A_1),plus_plus_int(C_2,B_1))) ) ).
fof(fact_65_add__right__mono,axiom,
! [C_1,A,B] :
( hBOOL(ord_less_eq_int(A,B))
=> hBOOL(ord_less_eq_int(plus_plus_int(A,C_1),plus_plus_int(B,C_1))) ) ).
fof(fact_66_add__le__cancel__left,axiom,
! [C,Aa,Ba] :
( hBOOL(ord_less_eq_int(plus_plus_int(C,Aa),plus_plus_int(C,Ba)))
<=> hBOOL(ord_less_eq_int(Aa,Ba)) ) ).
fof(fact_67_add__le__cancel__right,axiom,
! [Aa,C,Ba] :
( hBOOL(ord_less_eq_int(plus_plus_int(Aa,C),plus_plus_int(Ba,C)))
<=> hBOOL(ord_less_eq_int(Aa,Ba)) ) ).
fof(fact_68_order__refl,axiom,
! [X_1] : hBOOL(ord_less_eq_int(X_1,X_1)) ).
fof(fact_69_linorder__le__cases,axiom,
! [X,Y] :
( ~ hBOOL(ord_less_eq_int(X,Y))
=> hBOOL(ord_less_eq_int(Y,X)) ) ).
fof(fact_70_zle__refl,axiom,
! [W] : hBOOL(ord_less_eq_int(W,W)) ).
fof(fact_71_zle__linear,axiom,
! [Z,W] :
( hBOOL(ord_less_eq_int(Z,W))
| hBOOL(ord_less_eq_int(W,Z)) ) ).
fof(fact_72_zle__trans,axiom,
! [K,I,J] :
( hBOOL(ord_less_eq_int(I,J))
=> ( hBOOL(ord_less_eq_int(J,K))
=> hBOOL(ord_less_eq_int(I,K)) ) ) ).
fof(fact_73_zle__antisym,axiom,
! [Z,W] :
( hBOOL(ord_less_eq_int(Z,W))
=> ( hBOOL(ord_less_eq_int(W,Z))
=> Z = W ) ) ).
%----Conjectures (1)
fof(conj_0,conjecture,
times_times_int(twoSqu536811803sum2sq(product_Pair_int_int(a,b)),twoSqu536811803sum2sq(product_Pair_int_int(p,q))) = twoSqu536811803sum2sq(product_Pair_int_int(plus_plus_int(times_times_int(a,p),times_times_int(b,q)),minus_minus_int(times_times_int(a,q),times_times_int(b,p)))) ).
%------------------------------------------------------------------------------