TPTP Problem File: SWW649_2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SWW649_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : There and back again-T-WP parameter palindrome rec
% 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 : there_and_back_again-T-WP_parameter_palindrome_rec [Fil14]
% Status : Theorem
% Rating : 0.75 v8.2.0, 0.88 v8.1.0, 1.00 v7.5.0, 0.90 v7.4.0, 0.88 v7.3.0, 0.83 v7.1.0, 0.67 v7.0.0, 0.86 v6.4.0, 1.00 v6.3.0, 0.86 v6.2.0, 1.00 v6.1.0
% Syntax : Number of formulae : 98 ( 29 unt; 41 typ; 0 def)
% Number of atoms : 111 ( 57 equ)
% Maximal formula atoms : 13 ( 1 avg)
% Number of connectives : 59 ( 5 ~; 6 |; 11 &)
% ( 4 <=>; 33 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 5 avg)
% Maximal term depth : 6 ( 1 avg)
% Number arithmetic : 48 ( 14 atm; 9 fun; 13 num; 12 var)
% Number of types : 9 ( 7 usr; 1 ari)
% Number of type conns : 52 ( 25 >; 27 *; 0 +; 0 <<)
% Number of predicates : 6 ( 3 usr; 0 prp; 2-3 aty)
% Number of functors : 36 ( 31 usr; 11 con; 0-5 aty)
% Number of variables : 161 ( 157 !; 4 ?; 161 :)
% 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_sort1,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_sort1,axiom,
! [A: ty] : sort(list(A),nil(A)) ).
tff(cons,type,
cons: ( ty * uni * uni ) > uni ).
tff(cons_sort1,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_sort1,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_sort1,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_sort1,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_sort1,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(option,type,
option: ty > ty ).
tff(none,type,
none: ty > uni ).
tff(none_sort,axiom,
! [A: ty] : sort(option(A),none(A)) ).
tff(some,type,
some: ( ty * uni ) > uni ).
tff(some_sort,axiom,
! [A: ty,X: uni] : sort(option(A),some(A,X)) ).
tff(match_option,type,
match_option: ( ty * ty * uni * uni * uni ) > uni ).
tff(match_option_sort,axiom,
! [A: ty,A1: ty,X: uni,X1: uni,X2: uni] : sort(A1,match_option(A1,A,X,X1,X2)) ).
tff(match_option_None,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni] :
( sort(A1,Z)
=> ( match_option(A1,A,none(A),Z,Z1) = Z ) ) ).
tff(match_option_Some,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni,U: uni] :
( sort(A1,Z1)
=> ( match_option(A1,A,some(A,U),Z,Z1) = Z1 ) ) ).
tff(none_Some,axiom,
! [A: ty,V: uni] : none(A) != some(A,V) ).
tff(some_proj_1,type,
some_proj_1: ( ty * uni ) > uni ).
tff(some_proj_1_sort,axiom,
! [A: ty,X: uni] : sort(A,some_proj_1(A,X)) ).
tff(some_proj_1_def,axiom,
! [A: ty,U: uni] :
( sort(A,U)
=> ( some_proj_1(A,some(A,U)) = U ) ) ).
tff(option_inversion,axiom,
! [A: ty,U: uni] :
( sort(option(A),U)
=> ( ( U = none(A) )
| ( U = some(A,some_proj_1(A,U)) ) ) ) ).
tff(nth,type,
nth: ( ty * $int * uni ) > uni ).
tff(nth_sort,axiom,
! [A: ty,X: $int,X1: uni] : sort(option(A),nth(A,X,X1)) ).
tff(nth_def,axiom,
! [A: ty,N: $int] :
( ( nth(A,N,nil(A)) = none(A) )
& ! [X: uni,X1: uni] :
( ( ( N = 0 )
=> ( nth(A,N,cons(A,X,X1)) = some(A,X) ) )
& ( ( N != 0 )
=> ( nth(A,N,cons(A,X,X1)) = nth(A,$difference(N,1),X1) ) ) ) ) ).
tff(nth_none_1,axiom,
! [A: ty,L: uni,I: $int] :
( $less(I,0)
=> ( nth(A,I,L) = none(A) ) ) ).
tff(nth_none_2,axiom,
! [A: ty,L: uni,I: $int] :
( $lesseq(length(A,L),I)
=> ( nth(A,I,L) = none(A) ) ) ).
tff(nth_none_3,axiom,
! [A: ty,L: uni,I: $int] :
( ( nth(A,I,L) = none(A) )
=> ( $less(I,0)
| $lesseq(length(A,L),I) ) ) ).
tff(nth_append_1,axiom,
! [A: ty,L1: uni,L2: uni,I: $int] :
( $less(I,length(A,L1))
=> ( nth(A,I,infix_plpl(A,L1,L2)) = nth(A,I,L1) ) ) ).
tff(nth_append_2,axiom,
! [A: ty,L1: uni,L2: uni,I: $int] :
( $lesseq(length(A,L1),I)
=> ( nth(A,I,infix_plpl(A,L1,L2)) = nth(A,$difference(I,length(A,L1)),L2) ) ) ).
tff(pal,type,
pal: ( ty * uni * $int ) > $o ).
tff(pal_def,axiom,
! [A: ty,X: uni,N: $int] :
( pal(A,X,N)
<=> ! [I: $int] :
( ( $lesseq(0,I)
& $less(I,N) )
=> ( nth(A,I,X) = nth(A,$difference($difference(N,1),I),X) ) ) ) ).
tff(a,type,
a: $tType ).
tff(a1,type,
a1: ty ).
tff(t2tb,type,
t2tb: a > uni ).
tff(t2tb_sort4,axiom,
! [X: a] : sort(a1,t2tb(X)) ).
tff(tb2t,type,
tb2t: uni > a ).
tff(bridgeL,axiom,
! [I: a] : tb2t(t2tb(I)) = I ).
tff(bridgeR,axiom,
! [J: uni] :
( sort(a1,J)
=> ( t2tb(tb2t(J)) = J ) ) ).
tff(option_a1,type,
option_a1: $tType ).
tff(t2tb4,type,
t2tb4: option_a1 > uni ).
tff(t2tb_sort5,axiom,
! [X: option_a1] : sort(option(a1),t2tb4(X)) ).
tff(tb2t4,type,
tb2t4: uni > option_a1 ).
tff(bridgeL4,axiom,
! [I: option_a1] : tb2t4(t2tb4(I)) = I ).
tff(bridgeR4,axiom,
! [J: uni] :
( sort(option(a1),J)
=> ( t2tb4(tb2t4(J)) = J ) ) ).
tff(list_a1,type,
list_a1: $tType ).
tff(t2tb3,type,
t2tb3: list_a1 > uni ).
tff(t2tb_sort6,axiom,
! [X: list_a1] : sort(list(a1),t2tb3(X)) ).
tff(tb2t3,type,
tb2t3: uni > list_a1 ).
tff(bridgeL3,axiom,
! [I: list_a1] : tb2t3(t2tb3(I)) = I ).
tff(bridgeR3,axiom,
! [J: uni] : t2tb3(tb2t3(J)) = J ).
tff(wP_parameter_palindrome_rec,conjecture,
! [X: list_a1,Y: list_a1] :
( $lesseq(length(a1,t2tb3(Y)),length(a1,t2tb3(X)))
=> ! [X1: a,X2: list_a1] :
( ( Y = tb2t3(cons(a1,t2tb(X1),t2tb3(X2))) )
=> ! [X3: a,X4: list_a1] :
( ( X2 = tb2t3(cons(a1,t2tb(X3),t2tb3(X4))) )
=> ! [X5: a,X6: list_a1] :
( ( X = tb2t3(cons(a1,t2tb(X5),t2tb3(X6))) )
=> ( $lesseq(length(a1,t2tb3(X4)),length(a1,t2tb3(X6)))
=> ! [Result: list_a1] :
( ( ? [X0: list_a1] :
( ( length(a1,t2tb3(X0)) = length(a1,t2tb3(X4)) )
& ( X6 = tb2t3(infix_plpl(a1,t2tb3(X0),t2tb3(Result))) ) )
& pal(a1,t2tb3(X6),length(a1,t2tb3(X4))) )
=> ! [X11: a,Xs: list_a1] :
( ( Result = tb2t3(cons(a1,t2tb(X11),t2tb3(Xs))) )
=> ( ( tb2t4(nth(a1,$difference(length(a1,t2tb3(Y)),1),t2tb3(X))) = tb2t4(some(a1,t2tb(X11))) )
=> ( ( X5 = X11 )
=> ? [X0: list_a1] :
( ( length(a1,t2tb3(X0)) = length(a1,t2tb3(Y)) )
& ( X = tb2t3(infix_plpl(a1,t2tb3(X0),t2tb3(Xs))) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------