## TPTP Problem File: SWW591=2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SWW591=2 : TPTP v8.0.0. Released v6.1.0.
% Domain   : Software Verification
% Problem  : Fib memo-T-WP parameter memo fibo
% 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    : fib_memo-T-WP_parameter_memo_fibo [Fil14]

% Status   : Theorem
% Rating   : 0.38 v7.5.0, 0.40 v7.4.0, 0.25 v7.3.0, 0.17 v7.0.0, 0.14 v6.4.0, 0.00 v6.3.0, 0.29 v6.2.0, 0.38 v6.1.0
% Syntax   : Number of formulae    :   78 (  25 unt;  37 typ;   0 def)
%            Number of atoms       :   70 (  33 equ)
%            Maximal formula atoms :    9 (   0 avg)
%            Number of connectives :   32 (   3   ~;   2   |;   3   &)
%                                         (   1 <=>;  23  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   11 (   4 avg)
%            Maximal term depth    :    6 (   1 avg)
%            Number arithmetic     :   30 (   6 atm;   5 fun;  10 num;   9 var)
%            Number of types       :    8 (   6 usr;   1 ari)
%            Number of type conns  :   45 (  23   >;  22   *;   0   +;   0  <<)
%            Number of predicates  :    4 (   2 usr;   0 prp; 1-2 aty)
%            Number of functors    :   35 (  29 usr;  11 con; 0-5 aty)
%            Number of variables   :   94 (  94   !;   0   ?;  94   :)
% SPC      : TF0_THM_EQU_ARI

%------------------------------------------------------------------------------
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(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(compatOrderMult,axiom,
! [X: \$int,Y: \$int,Z: \$int] :
( \$lesseq(X,Y)
=> ( \$lesseq(0,Z)
=> \$lesseq(\$product(X,Z),\$product(Y,Z)) ) ) ).

tff(fib,type,
fib: \$int > \$int ).

tff(fib0,axiom,
fib(0) = 0 ).

tff(fib1,axiom,
fib(1) = 1 ).

tff(fibn,axiom,
! [N: \$int] :
( \$lesseq(2,N)
=> ( fib(N) = \$sum(fib(\$difference(N,1)),fib(\$difference(N,2))) ) ) ).

tff(ref,type,
ref: ty > ty ).

tff(mk_ref,type,
mk_ref: ( ty * uni ) > uni ).

tff(mk_ref_sort,axiom,
! [A: ty,X: uni] : sort(ref(A),mk_ref(A,X)) ).

tff(contents,type,
contents: ( ty * uni ) > uni ).

tff(contents_sort,axiom,
! [A: ty,X: uni] : sort(A,contents(A,X)) ).

tff(contents_def,axiom,
! [A: ty,U: uni] :
( sort(A,U)
=> ( contents(A,mk_ref(A,U)) = U ) ) ).

tff(ref_inversion,axiom,
! [A: ty,U: uni] :
( sort(ref(A),U)
=> ( U = mk_ref(A,contents(A,U)) ) ) ).

tff(map,type,
map: ( ty * ty ) > ty ).

tff(get,type,
get: ( ty * ty * uni * uni ) > uni ).

tff(get_sort,axiom,
! [A: ty,B: ty,X: uni,X1: uni] : sort(B,get(B,A,X,X1)) ).

tff(set,type,
set: ( ty * ty * uni * uni * uni ) > uni ).

tff(set_sort,axiom,
! [A: ty,B: ty,X: uni,X1: uni,X2: uni] : sort(map(A,B),set(B,A,X,X1,X2)) ).

tff(select_eq,axiom,
! [A: ty,B: ty,M: uni,A1: uni,A2: uni,B1: uni] :
( sort(B,B1)
=> ( ( A1 = A2 )
=> ( get(B,A,set(B,A,M,A1,B1),A2) = B1 ) ) ) ).

tff(select_neq,axiom,
! [A: ty,B: ty,M: uni,A1: uni,A2: uni] :
( sort(A,A1)
=> ( sort(A,A2)
=> ! [B1: uni] :
( ( A1 != A2 )
=> ( get(B,A,set(B,A,M,A1,B1),A2) = get(B,A,M,A2) ) ) ) ) ).

tff(const,type,
const: ( ty * ty * uni ) > uni ).

tff(const_sort,axiom,
! [A: ty,B: ty,X: uni] : sort(map(A,B),const(B,A,X)) ).

tff(const1,axiom,
! [A: ty,B: ty,B1: uni,A1: uni] :
( sort(B,B1)
=> ( get(B,A,const(B,A,B1),A1) = B1 ) ) ).

tff(map_int_lpoption_intrp,type,
map_int_lpoption_intrp: \$tType ).

tff(inv,type,
inv: map_int_lpoption_intrp > \$o ).

tff(t2tb,type,
t2tb: map_int_lpoption_intrp > uni ).

tff(t2tb_sort,axiom,
! [X: map_int_lpoption_intrp] : sort(map(int,option(int)),t2tb(X)) ).

tff(tb2t,type,
tb2t: uni > map_int_lpoption_intrp ).

tff(bridgeL,axiom,
! [I: map_int_lpoption_intrp] : ( tb2t(t2tb(I)) = I ) ).

tff(bridgeR,axiom,
! [J: uni] : ( t2tb(tb2t(J)) = J ) ).

tff(option_int,type,
option_int: \$tType ).

tff(t2tb1,type,
t2tb1: option_int > uni ).

tff(t2tb_sort1,axiom,
! [X: option_int] : sort(option(int),t2tb1(X)) ).

tff(tb2t1,type,
tb2t1: uni > option_int ).

tff(bridgeL1,axiom,
! [I: option_int] : ( tb2t1(t2tb1(I)) = I ) ).

tff(bridgeR1,axiom,
! [J: uni] : ( t2tb1(tb2t1(J)) = J ) ).

tff(t2tb2,type,
t2tb2: \$int > uni ).

tff(t2tb_sort2,axiom,
! [X: \$int] : sort(int,t2tb2(X)) ).

tff(tb2t2,type,
tb2t2: uni > \$int ).

tff(bridgeL2,axiom,
! [I: \$int] : ( tb2t2(t2tb2(I)) = I ) ).

tff(bridgeR2,axiom,
! [J: uni] : ( t2tb2(tb2t2(J)) = J ) ).

tff(inv_def,axiom,
! [T: map_int_lpoption_intrp] :
( inv(T)
<=> ! [X: \$int,Y: \$int] :
( ( tb2t1(get(option(int),int,t2tb(T),t2tb2(X))) = tb2t1(some(int,t2tb2(Y))) )
=> ( Y = fib(X) ) ) ) ).

tff(wP_parameter_memo_fibo,conjecture,
! [N: \$int,Table: map_int_lpoption_intrp] :
( ( \$lesseq(0,N)
& inv(Table) )
=> ( ( tb2t1(get(option(int),int,t2tb(Table),t2tb2(N))) = tb2t1(none(int)) )
=> ( ( \$lesseq(0,N)
& inv(Table) )
=> ! [Table1: map_int_lpoption_intrp] :
( inv(Table1)
=> ! [Table2: map_int_lpoption_intrp] :
( ( Table2 = tb2t(set(option(int),int,t2tb(Table1),t2tb2(N),some(int,t2tb2(fib(N))))) )
=> ( ( fib(N) = fib(N) )
& inv(Table2) ) ) ) ) ) ) ).

%------------------------------------------------------------------------------
```