TPTP Problem File: SWW624_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW624_2 : TPTP v9.0.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Mergesort list-T-permut prefix
% Version : Especial : Let and conditional terms encoded away.
% English :
% Refs : [Fil14] Filliatre (2014), Email to Geoff Sutcliffe
% : [BF+] Bobot et al. (URL), Toccata: Certified Programs and Cert
% Source : [Fil14]
% Names : mergesort_list-T-permut_prefix [Fil14]
% Status : Theorem
% Rating : 0.25 v9.0.0, 0.12 v8.2.0, 0.25 v7.5.0, 0.30 v7.4.0, 0.25 v7.3.0, 0.17 v7.1.0, 0.00 v7.0.0, 0.14 v6.4.0, 0.00 v6.3.0, 0.43 v6.2.0, 0.62 v6.1.0
% Syntax : Number of formulae : 142 ( 46 unt; 41 typ; 0 def)
% Number of atoms : 213 ( 71 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 122 ( 10 ~; 7 |; 33 &)
% ( 8 <=>; 64 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number arithmetic : 150 ( 43 atm; 22 fun; 46 num; 39 var)
% Number of types : 8 ( 6 usr; 1 ari)
% Number of type conns : 55 ( 26 >; 29 *; 0 +; 0 <<)
% Number of predicates : 8 ( 5 usr; 0 prp; 1-3 aty)
% Number of functors : 36 ( 30 usr; 11 con; 0-5 aty)
% Number of variables : 275 ( 269 !; 6 ?; 275 :)
% SPC : TF0_THM_EQU_ARI
% Comments :
%------------------------------------------------------------------------------
tff(uni,type,
uni: $tType ).
tff(ty,type,
ty: $tType ).
tff(sort,type,
sort: ( ty * uni ) > $o ).
tff(witness,type,
witness: ty > uni ).
tff(witness_sort,axiom,
! [A: ty] : sort(A,witness(A)) ).
tff(int,type,
int: ty ).
tff(real,type,
real: ty ).
tff(bool,type,
bool: $tType ).
tff(bool1,type,
bool1: ty ).
tff(true,type,
true: bool ).
tff(false,type,
false: bool ).
tff(match_bool,type,
match_bool: ( ty * bool * uni * uni ) > uni ).
tff(match_bool_sort,axiom,
! [A: ty,X: bool,X1: uni,X2: uni] : sort(A,match_bool(A,X,X1,X2)) ).
tff(match_bool_True,axiom,
! [A: ty,Z: uni,Z1: uni] :
( sort(A,Z)
=> ( match_bool(A,true,Z,Z1) = Z ) ) ).
tff(match_bool_False,axiom,
! [A: ty,Z: uni,Z1: uni] :
( sort(A,Z1)
=> ( match_bool(A,false,Z,Z1) = Z1 ) ) ).
tff(true_False,axiom,
true != false ).
tff(bool_inversion,axiom,
! [U: bool] :
( ( U = true )
| ( U = false ) ) ).
tff(tuple0,type,
tuple0: $tType ).
tff(tuple01,type,
tuple01: ty ).
tff(tuple02,type,
tuple02: tuple0 ).
tff(tuple0_inversion,axiom,
! [U: tuple0] : ( U = tuple02 ) ).
tff(qtmark,type,
qtmark: ty ).
tff(compatOrderMult,axiom,
! [X: $int,Y: $int,Z: $int] :
( $lesseq(X,Y)
=> ( $lesseq(0,Z)
=> $lesseq($product(X,Z),$product(Y,Z)) ) ) ).
tff(list,type,
list: ty > ty ).
tff(nil,type,
nil: ty > uni ).
tff(nil_sort,axiom,
! [A: ty] : sort(list(A),nil(A)) ).
tff(cons,type,
cons: ( ty * uni * uni ) > uni ).
tff(cons_sort,axiom,
! [A: ty,X: uni,X1: uni] : sort(list(A),cons(A,X,X1)) ).
tff(match_list,type,
match_list: ( ty * ty * uni * uni * uni ) > uni ).
tff(match_list_sort,axiom,
! [A: ty,A1: ty,X: uni,X1: uni,X2: uni] : sort(A1,match_list(A1,A,X,X1,X2)) ).
tff(match_list_Nil,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni] :
( sort(A1,Z)
=> ( match_list(A1,A,nil(A),Z,Z1) = Z ) ) ).
tff(match_list_Cons,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni,U: uni,U1: uni] :
( sort(A1,Z1)
=> ( match_list(A1,A,cons(A,U,U1),Z,Z1) = Z1 ) ) ).
tff(nil_Cons,axiom,
! [A: ty,V: uni,V1: uni] : ( nil(A) != cons(A,V,V1) ) ).
tff(cons_proj_1,type,
cons_proj_1: ( ty * uni ) > uni ).
tff(cons_proj_1_sort,axiom,
! [A: ty,X: uni] : sort(A,cons_proj_1(A,X)) ).
tff(cons_proj_1_def,axiom,
! [A: ty,U: uni,U1: uni] :
( sort(A,U)
=> ( cons_proj_1(A,cons(A,U,U1)) = U ) ) ).
tff(cons_proj_2,type,
cons_proj_2: ( ty * uni ) > uni ).
tff(cons_proj_2_sort,axiom,
! [A: ty,X: uni] : sort(list(A),cons_proj_2(A,X)) ).
tff(cons_proj_2_def,axiom,
! [A: ty,U: uni,U1: uni] : ( cons_proj_2(A,cons(A,U,U1)) = U1 ) ).
tff(list_inversion,axiom,
! [A: ty,U: uni] :
( ( U = nil(A) )
| ( U = cons(A,cons_proj_1(A,U),cons_proj_2(A,U)) ) ) ).
tff(length,type,
length: ( ty * uni ) > $int ).
tff(length_def,axiom,
! [A: ty] :
( ( length(A,nil(A)) = 0 )
& ! [X: uni,X1: uni] : ( length(A,cons(A,X,X1)) = $sum(1,length(A,X1)) ) ) ).
tff(length_nonnegative,axiom,
! [A: ty,L: uni] : $lesseq(0,length(A,L)) ).
tff(length_nil,axiom,
! [A: ty,L: uni] :
( ( length(A,L) = 0 )
<=> ( L = nil(A) ) ) ).
tff(infix_plpl,type,
infix_plpl: ( ty * uni * uni ) > uni ).
tff(infix_plpl_sort,axiom,
! [A: ty,X: uni,X1: uni] : sort(list(A),infix_plpl(A,X,X1)) ).
tff(infix_plpl_def,axiom,
! [A: ty,L2: uni] :
( ( infix_plpl(A,nil(A),L2) = L2 )
& ! [X: uni,X1: uni] : ( infix_plpl(A,cons(A,X,X1),L2) = cons(A,X,infix_plpl(A,X1,L2)) ) ) ).
tff(append_assoc,axiom,
! [A: ty,L1: uni,L2: uni,L3: uni] : ( infix_plpl(A,L1,infix_plpl(A,L2,L3)) = infix_plpl(A,infix_plpl(A,L1,L2),L3) ) ).
tff(append_l_nil,axiom,
! [A: ty,L: uni] : ( infix_plpl(A,L,nil(A)) = L ) ).
tff(append_length,axiom,
! [A: ty,L1: uni,L2: uni] : ( length(A,infix_plpl(A,L1,L2)) = $sum(length(A,L1),length(A,L2)) ) ).
tff(mem,type,
mem: ( ty * uni * uni ) > $o ).
tff(mem_def,axiom,
! [A: ty,X: uni] :
( sort(A,X)
=> ( ~ mem(A,X,nil(A))
& ! [X1: uni,X2: uni] :
( sort(A,X1)
=> ( mem(A,X,cons(A,X1,X2))
<=> ( ( X = X1 )
| mem(A,X,X2) ) ) ) ) ) ).
tff(mem_append,axiom,
! [A: ty,X: uni,L1: uni,L2: uni] :
( mem(A,X,infix_plpl(A,L1,L2))
<=> ( mem(A,X,L1)
| mem(A,X,L2) ) ) ).
tff(mem_decomp,axiom,
! [A: ty,X: uni,L: uni] :
( mem(A,X,L)
=> ? [L1: uni,L2: uni] :
( sort(list(A),L1)
& sort(list(A),L2)
& ( L = infix_plpl(A,L1,cons(A,X,L2)) ) ) ) ).
tff(num_occ,type,
num_occ: ( ty * uni * uni ) > $int ).
tff(num_occ_def,axiom,
! [A: ty,X: uni] :
( sort(A,X)
=> ( ( num_occ(A,X,nil(A)) = 0 )
& ! [X1: uni,X2: uni] :
( sort(A,X1)
=> ( ( ( X = X1 )
=> ( num_occ(A,X,cons(A,X1,X2)) = $sum(1,num_occ(A,X,X2)) ) )
& ( ( X != X1 )
=> ( num_occ(A,X,cons(A,X1,X2)) = $sum(0,num_occ(A,X,X2)) ) ) ) ) ) ) ).
tff(mem_Num_Occ,axiom,
! [A: ty,X: uni,L: uni] :
( mem(A,X,L)
<=> $less(0,num_occ(A,X,L)) ) ).
tff(append_Num_Occ,axiom,
! [A: ty,X: uni,L1: uni,L2: uni] : ( num_occ(A,X,infix_plpl(A,L1,L2)) = $sum(num_occ(A,X,L1),num_occ(A,X,L2)) ) ).
tff(reverse,type,
reverse: ( ty * uni ) > uni ).
tff(reverse_sort,axiom,
! [A: ty,X: uni] : sort(list(A),reverse(A,X)) ).
tff(reverse_def,axiom,
! [A: ty] :
( ( reverse(A,nil(A)) = nil(A) )
& ! [X: uni,X1: uni] : ( reverse(A,cons(A,X,X1)) = infix_plpl(A,reverse(A,X1),cons(A,X,nil(A))) ) ) ).
tff(reverse_append,axiom,
! [A: ty,L1: uni,L2: uni,X: uni] : ( infix_plpl(A,reverse(A,cons(A,X,L1)),L2) = infix_plpl(A,reverse(A,L1),cons(A,X,L2)) ) ).
tff(reverse_cons,axiom,
! [A: ty,L: uni,X: uni] : ( reverse(A,cons(A,X,L)) = infix_plpl(A,reverse(A,L),cons(A,X,nil(A))) ) ).
tff(reverse_reverse,axiom,
! [A: ty,L: uni] : ( reverse(A,reverse(A,L)) = L ) ).
tff(reverse_mem,axiom,
! [A: ty,L: uni,X: uni] :
( mem(A,X,L)
<=> mem(A,X,reverse(A,L)) ) ).
tff(reverse_length,axiom,
! [A: ty,L: uni] : ( length(A,reverse(A,L)) = length(A,L) ) ).
tff(reverse_num_occ,axiom,
! [A: ty,X: uni,L: uni] : ( num_occ(A,X,L) = num_occ(A,X,reverse(A,L)) ) ).
tff(permut,type,
permut: ( ty * uni * uni ) > $o ).
tff(permut_def,axiom,
! [A: ty,L1: uni,L2: uni] :
( ( permut(A,L1,L2)
=> ! [X: uni] : ( num_occ(A,X,L1) = num_occ(A,X,L2) ) )
& ( ! [X: uni] :
( sort(A,X)
=> ( num_occ(A,X,L1) = num_occ(A,X,L2) ) )
=> permut(A,L1,L2) ) ) ).
tff(permut_refl,axiom,
! [A: ty,L: uni] : permut(A,L,L) ).
tff(permut_sym,axiom,
! [A: ty,L1: uni,L2: uni] :
( permut(A,L1,L2)
=> permut(A,L2,L1) ) ).
tff(permut_trans,axiom,
! [A: ty,L1: uni,L2: uni,L3: uni] :
( permut(A,L1,L2)
=> ( permut(A,L2,L3)
=> permut(A,L1,L3) ) ) ).
tff(permut_cons,axiom,
! [A: ty,X: uni,L1: uni,L2: uni] :
( permut(A,L1,L2)
=> permut(A,cons(A,X,L1),cons(A,X,L2)) ) ).
tff(permut_swap,axiom,
! [A: ty,X: uni,Y: uni,L: uni] : permut(A,cons(A,X,cons(A,Y,L)),cons(A,Y,cons(A,X,L))) ).
tff(permut_cons_append,axiom,
! [A: ty,X: uni,L1: uni,L2: uni] : permut(A,infix_plpl(A,cons(A,X,L1),L2),infix_plpl(A,L1,cons(A,X,L2))) ).
tff(permut_assoc,axiom,
! [A: ty,L1: uni,L2: uni,L3: uni] : permut(A,infix_plpl(A,infix_plpl(A,L1,L2),L3),infix_plpl(A,L1,infix_plpl(A,L2,L3))) ).
tff(permut_append,axiom,
! [A: ty,L1: uni,L2: uni,K1: uni,K2: uni] :
( permut(A,L1,K1)
=> ( permut(A,L2,K2)
=> permut(A,infix_plpl(A,L1,L2),infix_plpl(A,K1,K2)) ) ) ).
tff(permut_append_swap,axiom,
! [A: ty,L1: uni,L2: uni] : permut(A,infix_plpl(A,L1,L2),infix_plpl(A,L2,L1)) ).
tff(permut_mem,axiom,
! [A: ty,X: uni,L1: uni,L2: uni] :
( permut(A,L1,L2)
=> ( mem(A,X,L1)
=> mem(A,X,L2) ) ) ).
tff(permut_length,axiom,
! [A: ty,L1: uni,L2: uni] :
( permut(A,L1,L2)
=> ( length(A,L1) = length(A,L2) ) ) ).
tff(elt,type,
elt: $tType ).
tff(elt1,type,
elt1: ty ).
tff(le,type,
le: ( elt * elt ) > $o ).
tff(refl,axiom,
! [X: elt] : le(X,X) ).
tff(trans,axiom,
! [X: elt,Y: elt,Z: elt] :
( le(X,Y)
=> ( le(Y,Z)
=> le(X,Z) ) ) ).
tff(total,axiom,
! [X: elt,Y: elt] :
( le(X,Y)
| le(Y,X) ) ).
tff(list_elt,type,
list_elt: $tType ).
tff(sorted,type,
sorted: list_elt > $o ).
tff(t2tb,type,
t2tb: list_elt > uni ).
tff(t2tb_sort,axiom,
! [X: list_elt] : sort(list(elt1),t2tb(X)) ).
tff(tb2t,type,
tb2t: uni > list_elt ).
tff(bridgeL,axiom,
! [I: list_elt] : ( tb2t(t2tb(I)) = I ) ).
tff(bridgeR,axiom,
! [J: uni] : ( t2tb(tb2t(J)) = J ) ).
tff(sorted_Nil,axiom,
sorted(tb2t(nil(elt1))) ).
tff(t2tb1,type,
t2tb1: elt > uni ).
tff(t2tb_sort1,axiom,
! [X: elt] : sort(elt1,t2tb1(X)) ).
tff(tb2t1,type,
tb2t1: uni > elt ).
tff(bridgeL1,axiom,
! [I: elt] : ( tb2t1(t2tb1(I)) = I ) ).
tff(bridgeR1,axiom,
! [J: uni] :
( sort(elt1,J)
=> ( t2tb1(tb2t1(J)) = J ) ) ).
tff(sorted_One,axiom,
! [X: elt] : sorted(tb2t(cons(elt1,t2tb1(X),nil(elt1)))) ).
tff(sorted_Two,axiom,
! [X: elt,Y: elt,L: list_elt] :
( le(X,Y)
=> ( sorted(tb2t(cons(elt1,t2tb1(Y),t2tb(L))))
=> sorted(tb2t(cons(elt1,t2tb1(X),cons(elt1,t2tb1(Y),t2tb(L))))) ) ) ).
tff(sorted_inversion,axiom,
! [Z: list_elt] :
( sorted(Z)
=> ( ( Z = tb2t(nil(elt1)) )
| ? [X: elt] : ( Z = tb2t(cons(elt1,t2tb1(X),nil(elt1))) )
| ? [X: elt,Y: elt,L: list_elt] :
( le(X,Y)
& sorted(tb2t(cons(elt1,t2tb1(Y),t2tb(L))))
& ( Z = tb2t(cons(elt1,t2tb1(X),cons(elt1,t2tb1(Y),t2tb(L)))) ) ) ) ) ).
tff(sorted_mem,axiom,
! [X: elt,L: list_elt] :
( ( ! [Y: elt] :
( mem(elt1,t2tb1(Y),t2tb(L))
=> le(X,Y) )
& sorted(L) )
<=> sorted(tb2t(cons(elt1,t2tb1(X),t2tb(L)))) ) ).
tff(sorted_append,axiom,
! [L1: list_elt,L2: list_elt] :
( ( sorted(L1)
& sorted(L2)
& ! [X: elt,Y: elt] :
( mem(elt1,t2tb1(X),t2tb(L1))
=> ( mem(elt1,t2tb1(Y),t2tb(L2))
=> le(X,Y) ) ) )
<=> sorted(tb2t(infix_plpl(elt1,t2tb(L1),t2tb(L2)))) ) ).
tff(rev_append,type,
rev_append: ( ty * uni * uni ) > uni ).
tff(rev_append_sort,axiom,
! [A: ty,X: uni,X1: uni] : sort(list(A),rev_append(A,X,X1)) ).
tff(rev_append_def,axiom,
! [A: ty,T: uni] :
( ( rev_append(A,nil(A),T) = T )
& ! [X: uni,R: uni] : ( rev_append(A,cons(A,X,R),T) = rev_append(A,R,cons(A,X,T)) ) ) ).
tff(rev_append_append_l,axiom,
! [A: ty,R: uni,S: uni,T: uni] : ( rev_append(A,infix_plpl(A,R,S),T) = rev_append(A,S,rev_append(A,R,T)) ) ).
tff(rev_append_append_r,axiom,
! [A: ty,R: uni,S: uni,T: uni] : ( rev_append(A,R,infix_plpl(A,S,T)) = rev_append(A,rev_append(A,S,R),T) ) ).
tff(rev_append_length,axiom,
! [A: ty,S: uni,T: uni] : ( length(A,rev_append(A,S,T)) = $sum(length(A,S),length(A,T)) ) ).
tff(rev_append_def1,axiom,
! [A: ty,R: uni,S: uni] : ( rev_append(A,R,S) = infix_plpl(A,reverse(A,R),S) ) ).
tff(sorted_reverse_cons,axiom,
! [Acc: list_elt,X1: elt] :
( sorted(tb2t(reverse(elt1,t2tb(Acc))))
=> ( ! [X: elt] :
( mem(elt1,t2tb1(X),t2tb(Acc))
=> le(X,X1) )
=> sorted(tb2t(reverse(elt1,cons(elt1,t2tb1(X1),t2tb(Acc))))) ) ) ).
tff(sorted_rev_append,axiom,
! [Acc: list_elt,L: list_elt] :
( sorted(tb2t(reverse(elt1,t2tb(Acc))))
=> ( sorted(L)
=> ( ! [X: elt,Y: elt] :
( mem(elt1,t2tb1(X),t2tb(Acc))
=> ( mem(elt1,t2tb1(Y),t2tb(L))
=> le(X,Y) ) )
=> sorted(tb2t(reverse(elt1,rev_append(elt1,t2tb(L),t2tb(Acc))))) ) ) ) ).
tff(sorted_reverse_mem,axiom,
! [X: elt,L: list_elt] :
( sorted(tb2t(reverse(elt1,cons(elt1,t2tb1(X),t2tb(L)))))
=> ! [Y: elt] :
( mem(elt1,t2tb1(Y),t2tb(L))
=> le(Y,X) ) ) ).
tff(sorted_reverse_cons2,axiom,
! [X: elt,L: list_elt] :
( sorted(tb2t(reverse(elt1,cons(elt1,t2tb1(X),t2tb(L)))))
=> sorted(tb2t(reverse(elt1,t2tb(L)))) ) ).
tff(prefix,type,
prefix: ( ty * $int * uni ) > uni ).
tff(prefix_sort,axiom,
! [A: ty,X: $int,X1: uni] : sort(list(A),prefix(A,X,X1)) ).
tff(prefix_def1,axiom,
! [A: ty,L: uni] : ( prefix(A,0,L) = nil(A) ) ).
tff(prefix_def2,axiom,
! [A: ty,N: $int,X: uni,L: uni] :
( $less(0,N)
=> ( prefix(A,N,cons(A,X,L)) = cons(A,X,prefix(A,$difference(N,1),L)) ) ) ).
tff(prefix_length,axiom,
! [A: ty,N: $int,L: uni] :
( ( $lesseq(0,N)
& $lesseq(N,length(A,L)) )
=> ( length(A,prefix(A,N,L)) = N ) ) ).
tff(prefix_append,axiom,
! [A: ty,N: $int,L1: uni,L2: uni] :
( ( $lesseq(length(A,L1),N)
& $lesseq(N,$sum(length(A,L1),length(A,L2))) )
=> ( prefix(A,N,infix_plpl(A,L1,L2)) = infix_plpl(A,prefix(A,length(A,L1),L1),prefix(A,$difference(N,length(A,L1)),L2)) ) ) ).
tff(abs,type,
abs: $int > $int ).
tff(abs_def,axiom,
! [X: $int] :
( ( $lesseq(0,X)
=> ( abs(X) = X ) )
& ( ~ $lesseq(0,X)
=> ( abs(X) = $uminus(X) ) ) ) ).
tff(abs_le,axiom,
! [X: $int,Y: $int] :
( $lesseq(abs(X),Y)
<=> ( $lesseq($uminus(Y),X)
& $lesseq(X,Y) ) ) ).
tff(abs_pos,axiom,
! [X: $int] : $lesseq(0,abs(X)) ).
tff(div,type,
div: ( $int * $int ) > $int ).
tff(mod,type,
mod: ( $int * $int ) > $int ).
tff(div_mod,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> ( X = $sum($product(Y,div(X,Y)),mod(X,Y)) ) ) ).
tff(div_bound,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(0,Y) )
=> ( $lesseq(0,div(X,Y))
& $lesseq(div(X,Y),X) ) ) ).
tff(mod_bound,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> ( $less($uminus(abs(Y)),mod(X,Y))
& $less(mod(X,Y),abs(Y)) ) ) ).
tff(div_sign_pos,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(0,Y) )
=> $lesseq(0,div(X,Y)) ) ).
tff(div_sign_neg,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
& $less(0,Y) )
=> $lesseq(div(X,Y),0) ) ).
tff(mod_sign_pos,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& ( Y != 0 ) )
=> $lesseq(0,mod(X,Y)) ) ).
tff(mod_sign_neg,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
& ( Y != 0 ) )
=> $lesseq(mod(X,Y),0) ) ).
tff(rounds_toward_zero,axiom,
! [X: $int,Y: $int] :
( ( Y != 0 )
=> $lesseq(abs($product(div(X,Y),Y)),abs(X)) ) ).
tff(div_1,axiom,
! [X: $int] : ( div(X,1) = X ) ).
tff(mod_1,axiom,
! [X: $int] : ( mod(X,1) = 0 ) ).
tff(div_inf,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(X,Y) )
=> ( div(X,Y) = 0 ) ) ).
tff(mod_inf,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(0,X)
& $less(X,Y) )
=> ( mod(X,Y) = X ) ) ).
tff(div_mult,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $less(0,X)
& $lesseq(0,Y)
& $lesseq(0,Z) )
=> ( div($sum($product(X,Y),Z),X) = $sum(Y,div(Z,X)) ) ) ).
tff(mod_mult,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $less(0,X)
& $lesseq(0,Y)
& $lesseq(0,Z) )
=> ( mod($sum($product(X,Y),Z),X) = mod(Z,X) ) ) ).
tff(permut_prefix,conjecture,
( permut(elt1,prefix(elt1,length(elt1,nil(elt1)),nil(elt1)),nil(elt1))
& ! [X: elt,X1: list_elt] :
( permut(elt1,prefix(elt1,length(elt1,t2tb(X1)),t2tb(X1)),t2tb(X1))
=> permut(elt1,prefix(elt1,length(elt1,cons(elt1,t2tb1(X),t2tb(X1))),cons(elt1,t2tb1(X),t2tb(X1))),cons(elt1,t2tb1(X),t2tb(X1))) ) ) ).
%------------------------------------------------------------------------------