## TPTP Problem File: SWW634=2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SWW634=2 : TPTP v8.0.0. Released v6.1.0.
% Domain   : Software Verification
% Problem  : Queens-T-WP parameter queens3
% 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    : queens-T-WP_parameter_queens3 [Fil14]

% Status   : Theorem
% Rating   : 0.25 v7.5.0, 0.40 v7.4.0, 0.38 v7.3.0, 0.17 v7.0.0, 0.43 v6.4.0, 0.33 v6.3.0, 0.57 v6.2.0, 0.75 v6.1.0
% Syntax   : Number of formulae    :  132 (  34 unt;  56 typ;   0 def)
%            Number of atoms       :  215 (  46 equ)
%            Maximal formula atoms :   40 (   1 avg)
%            Number of connectives :  161 (  22   ~;   3   |;  50   &)
%                                         (  21 <=>;  65  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   5 avg)
%            Maximal term depth    :    5 (   1 avg)
%            Number arithmetic     :  139 (  48 atm;  18 fun;  33 num;  40 var)
%            Number of types       :    9 (   7 usr;   1 ari)
%            Number of type conns  :   82 (  40   >;  42   *;   0   +;   0  <<)
%            Number of predicates  :   13 (   9 usr;   1 prp; 0-4 aty)
%            Number of functors    :   45 (  40 usr;  11 con; 0-5 aty)
%            Number of variables   :  215 ( 212   !;   3   ?; 215   :)
% SPC      : TF0_THM_EQU_ARI

%------------------------------------------------------------------------------
tff(uni,type,
uni: \$tType ).

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

tff(sort,type,
sort1: ( ty * uni ) > \$o ).

tff(witness,type,
witness1: ty > uni ).

tff(witness_sort1,axiom,
! [A: ty] : sort1(A,witness1(A)) ).

tff(int,type,
int: ty ).

tff(real,type,
real: ty ).

tff(bool,type,
bool1: \$tType ).

tff(bool1,type,
bool: ty ).

tff(true,type,
true1: bool1 ).

tff(false,type,
false1: bool1 ).

tff(match_bool,type,
match_bool1: ( ty * bool1 * uni * uni ) > uni ).

tff(match_bool_sort1,axiom,
! [A: ty,X: bool1,X1: uni,X2: uni] : sort1(A,match_bool1(A,X,X1,X2)) ).

tff(match_bool_True,axiom,
! [A: ty,Z: uni,Z1: uni] :
( sort1(A,Z)
=> ( match_bool1(A,true1,Z,Z1) = Z ) ) ).

tff(match_bool_False,axiom,
! [A: ty,Z: uni,Z1: uni] :
( sort1(A,Z1)
=> ( match_bool1(A,false1,Z,Z1) = Z1 ) ) ).

tff(true_False,axiom,
true1 != false1 ).

tff(bool_inversion,axiom,
! [U: bool1] :
( ( U = true1 )
| ( U = false1 ) ) ).

tff(tuple0,type,
tuple02: \$tType ).

tff(tuple01,type,
tuple0: ty ).

tff(tuple02,type,
tuple03: tuple02 ).

tff(tuple0_inversion,axiom,
! [U: tuple02] : ( U = tuple03 ) ).

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(set,type,
set: ty > ty ).

tff(mem,type,
mem: ( ty * uni * uni ) > \$o ).

tff(infix_eqeq,type,
infix_eqeq: ( ty * uni * uni ) > \$o ).

tff(infix_eqeq_def,axiom,
! [A: ty,S1: uni,S2: uni] :
( ( infix_eqeq(A,S1,S2)
=> ! [X: uni] :
( mem(A,X,S1)
<=> mem(A,X,S2) ) )
& ( ! [X: uni] :
( sort1(A,X)
=> ( mem(A,X,S1)
<=> mem(A,X,S2) ) )
=> infix_eqeq(A,S1,S2) ) ) ).

tff(extensionality,axiom,
! [A: ty,S1: uni,S2: uni] :
( sort1(set(A),S1)
=> ( sort1(set(A),S2)
=> ( infix_eqeq(A,S1,S2)
=> ( S1 = S2 ) ) ) ) ).

tff(subset,type,
subset: ( ty * uni * uni ) > \$o ).

tff(subset_def,axiom,
! [A: ty,S1: uni,S2: uni] :
( ( subset(A,S1,S2)
=> ! [X: uni] :
( mem(A,X,S1)
=> mem(A,X,S2) ) )
& ( ! [X: uni] :
( sort1(A,X)
=> ( mem(A,X,S1)
=> mem(A,X,S2) ) )
=> subset(A,S1,S2) ) ) ).

tff(subset_refl,axiom,
! [A: ty,S: uni] : subset(A,S,S) ).

tff(subset_trans,axiom,
! [A: ty,S1: uni,S2: uni,S3: uni] :
( subset(A,S1,S2)
=> ( subset(A,S2,S3)
=> subset(A,S1,S3) ) ) ).

tff(empty,type,
empty: ty > uni ).

tff(empty_sort1,axiom,
! [A: ty] : sort1(set(A),empty(A)) ).

tff(is_empty,type,
is_empty: ( ty * uni ) > \$o ).

tff(is_empty_def,axiom,
! [A: ty,S: uni] :
( ( is_empty(A,S)
=> ! [X: uni] : ~ mem(A,X,S) )
& ( ! [X: uni] :
( sort1(A,X)
=> ~ mem(A,X,S) )
=> is_empty(A,S) ) ) ).

tff(empty_def1,axiom,
! [A: ty] : is_empty(A,empty(A)) ).

tff(mem_empty,axiom,
! [A: ty,X: uni] :
( mem(A,X,empty(A))
<=> \$false ) ).

add: ( ty * uni * uni ) > uni ).

! [A: ty,X: uni,X1: uni] : sort1(set(A),add(A,X,X1)) ).

! [A: ty,X: uni,Y: uni] :
( sort1(A,X)
=> ( sort1(A,Y)
=> ! [S: uni] :
<=> ( ( X = Y )
| mem(A,X,S) ) ) ) ) ).

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

tff(remove_sort1,axiom,
! [A: ty,X: uni,X1: uni] : sort1(set(A),remove(A,X,X1)) ).

tff(remove_def1,axiom,
! [A: ty,X: uni,Y: uni,S: uni] :
( sort1(A,X)
=> ( sort1(A,Y)
=> ( mem(A,X,remove(A,Y,S))
<=> ( ( X != Y )
& mem(A,X,S) ) ) ) ) ).

! [A: ty,X: uni,S: uni] :
( sort1(set(A),S)
=> ( mem(A,X,S)
=> ( add(A,X,remove(A,X,S)) = S ) ) ) ).

! [A: ty,X: uni,S: uni] : ( remove(A,X,add(A,X,S)) = remove(A,X,S) ) ).

tff(subset_remove,axiom,
! [A: ty,X: uni,S: uni] : subset(A,remove(A,X,S),S) ).

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

tff(union_sort1,axiom,
! [A: ty,X: uni,X1: uni] : sort1(set(A),union(A,X,X1)) ).

tff(union_def1,axiom,
! [A: ty,S1: uni,S2: uni,X: uni] :
( mem(A,X,union(A,S1,S2))
<=> ( mem(A,X,S1)
| mem(A,X,S2) ) ) ).

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

tff(inter_sort1,axiom,
! [A: ty,X: uni,X1: uni] : sort1(set(A),inter(A,X,X1)) ).

tff(inter_def1,axiom,
! [A: ty,S1: uni,S2: uni,X: uni] :
( mem(A,X,inter(A,S1,S2))
<=> ( mem(A,X,S1)
& mem(A,X,S2) ) ) ).

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

tff(diff_sort1,axiom,
! [A: ty,X: uni,X1: uni] : sort1(set(A),diff(A,X,X1)) ).

tff(diff_def1,axiom,
! [A: ty,S1: uni,S2: uni,X: uni] :
( mem(A,X,diff(A,S1,S2))
<=> ( mem(A,X,S1)
& ~ mem(A,X,S2) ) ) ).

tff(subset_diff,axiom,
! [A: ty,S1: uni,S2: uni] : subset(A,diff(A,S1,S2),S1) ).

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

tff(choose_sort1,axiom,
! [A: ty,X: uni] : sort1(A,choose(A,X)) ).

tff(choose_def,axiom,
! [A: ty,S: uni] :
( ~ is_empty(A,S)
=> mem(A,choose(A,S),S) ) ).

tff(cardinal,type,
cardinal1: ( ty * uni ) > \$int ).

tff(cardinal_nonneg,axiom,
! [A: ty,S: uni] : \$lesseq(0,cardinal1(A,S)) ).

tff(cardinal_empty,axiom,
! [A: ty,S: uni] :
( ( cardinal1(A,S) = 0 )
<=> is_empty(A,S) ) ).

! [A: ty,X: uni,S: uni] :
( ~ mem(A,X,S)
=> ( cardinal1(A,add(A,X,S)) = \$sum(1,cardinal1(A,S)) ) ) ).

tff(cardinal_remove,axiom,
! [A: ty,X: uni,S: uni] :
( mem(A,X,S)
=> ( cardinal1(A,S) = \$sum(1,cardinal1(A,remove(A,X,S))) ) ) ).

tff(cardinal_subset,axiom,
! [A: ty,S1: uni,S2: uni] :
( subset(A,S1,S2)
=> \$lesseq(cardinal1(A,S1),cardinal1(A,S2)) ) ).

tff(cardinal1,axiom,
! [A: ty,S: uni] :
( ( cardinal1(A,S) = 1 )
=> ! [X: uni] :
( sort1(A,X)
=> ( mem(A,X,S)
=> ( X = choose(A,S) ) ) ) ) ).

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

tff(min_elt,type,
min_elt1: set_int > \$int ).

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

tff(t2tb_sort,axiom,
! [X: set_int] : sort1(set(int),t2tb(X)) ).

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

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

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

tff(t2tb1,type,
t2tb1: \$int > uni ).

tff(t2tb_sort1,axiom,
! [X: \$int] : sort1(int,t2tb1(X)) ).

tff(tb2t1,type,
tb2t1: uni > \$int ).

tff(bridgeL1,axiom,
! [I: \$int] : ( tb2t1(t2tb1(I)) = I ) ).

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

tff(min_elt_def1,axiom,
! [S: set_int] :
( ~ is_empty(int,t2tb(S))
=> mem(int,t2tb1(min_elt1(S)),t2tb(S)) ) ).

tff(min_elt_def2,axiom,
! [S: set_int] :
( ~ is_empty(int,t2tb(S))
=> ! [X: \$int] :
( mem(int,t2tb1(X),t2tb(S))
=> \$lesseq(min_elt1(S),X) ) ) ).

tff(max_elt,type,
max_elt1: set_int > \$int ).

tff(max_elt_def1,axiom,
! [S: set_int] :
( ~ is_empty(int,t2tb(S))
=> mem(int,t2tb1(max_elt1(S)),t2tb(S)) ) ).

tff(max_elt_def2,axiom,
! [S: set_int] :
( ~ is_empty(int,t2tb(S))
=> ! [X: \$int] :
( mem(int,t2tb1(X),t2tb(S))
=> \$lesseq(X,max_elt1(S)) ) ) ).

tff(below,type,
below1: \$int > set_int ).

tff(below_def,axiom,
! [X: \$int,N: \$int] :
( mem(int,t2tb1(X),t2tb(below1(N)))
<=> ( \$lesseq(0,X)
& \$less(X,N) ) ) ).

tff(cardinal_below,axiom,
! [N: \$int] :
( ( \$lesseq(0,N)
=> ( cardinal1(int,t2tb(below1(N))) = N ) )
& ( ~ \$lesseq(0,N)
=> ( cardinal1(int,t2tb(below1(N))) = 0 ) ) ) ).

tff(succ,type,
succ1: set_int > set_int ).

tff(succ_def,axiom,
! [S: set_int,I: \$int] :
( mem(int,t2tb1(I),t2tb(succ1(S)))
<=> ( \$lesseq(1,I)
& mem(int,t2tb1(\$difference(I,1)),t2tb(S)) ) ) ).

tff(pred,type,
pred1: set_int > set_int ).

tff(pred_def,axiom,
! [S: set_int,I: \$int] :
( mem(int,t2tb1(I),t2tb(pred1(S)))
<=> ( \$lesseq(0,I)
& mem(int,t2tb1(\$sum(I,1)),t2tb(S)) ) ) ).

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

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

tff(mk_ref_sort1,axiom,
! [A: ty,X: uni] : sort1(ref(A),mk_ref(A,X)) ).

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

tff(contents_sort1,axiom,
! [A: ty,X: uni] : sort1(A,contents(A,X)) ).

tff(contents_def1,axiom,
! [A: ty,U: uni] :
( sort1(A,U)
=> ( contents(A,mk_ref(A,U)) = U ) ) ).

tff(ref_inversion1,axiom,
! [A: ty,U: uni] :
( sort1(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_sort1,axiom,
! [A: ty,B: ty,X: uni,X1: uni] : sort1(B,get(B,A,X,X1)) ).

tff(set1,type,
set1: ( ty * ty * uni * uni * uni ) > uni ).

tff(set_sort1,axiom,
! [A: ty,B: ty,X: uni,X1: uni,X2: uni] : sort1(map(A,B),set1(B,A,X,X1,X2)) ).

tff(select_eq,axiom,
! [A: ty,B: ty,M: uni,A1: uni,A2: uni,B1: uni] :
( sort1(B,B1)
=> ( ( A1 = A2 )
=> ( get(B,A,set1(B,A,M,A1,B1),A2) = B1 ) ) ) ).

tff(select_neq,axiom,
! [A: ty,B: ty,M: uni,A1: uni,A2: uni] :
( sort1(A,A1)
=> ( sort1(A,A2)
=> ! [B1: uni] :
( ( A1 != A2 )
=> ( get(B,A,set1(B,A,M,A1,B1),A2) = get(B,A,M,A2) ) ) ) ) ).

tff(const1,type,
const: ( ty * ty * uni ) > uni ).

tff(const_sort1,axiom,
! [A: ty,B: ty,X: uni] : sort1(map(A,B),const(B,A,X)) ).

tff(const,axiom,
! [A: ty,B: ty,B1: uni,A1: uni] :
( sort1(B,B1)
=> ( get(B,A,const(B,A,B1),A1) = B1 ) ) ).

tff(n,type,
n1: \$int ).

tff(eq_prefix,type,
eq_prefix1: ( ty * uni * uni * \$int ) > \$o ).

tff(eq_prefix_def,axiom,
! [A: ty,T: uni,U: uni,I: \$int] :
( eq_prefix1(A,T,U,I)
<=> ! [K: \$int] :
( ( \$lesseq(0,K)
& \$less(K,I) )
=> ( get(A,int,T,t2tb1(K)) = get(A,int,U,t2tb1(K)) ) ) ) ).

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

tff(partial_solution,type,
partial_solution1: ( \$int * map_int_int ) > \$o ).

tff(t2tb2,type,
t2tb2: map_int_int > uni ).

tff(t2tb_sort2,axiom,
! [X: map_int_int] : sort1(map(int,int),t2tb2(X)) ).

tff(tb2t2,type,
tb2t2: uni > map_int_int ).

tff(bridgeL2,axiom,
! [I: map_int_int] : ( tb2t2(t2tb2(I)) = I ) ).

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

tff(partial_solution_def,axiom,
! [K: \$int,S: map_int_int] :
( partial_solution1(K,S)
<=> ! [I: \$int] :
( ( \$lesseq(0,I)
& \$less(I,K) )
=> ( \$lesseq(0,tb2t1(get(int,int,t2tb2(S),t2tb1(I))))
& \$less(tb2t1(get(int,int,t2tb2(S),t2tb1(I))),n1)
& ! [J: \$int] :
( ( \$lesseq(0,J)
& \$less(J,I) )
=> ( ( tb2t1(get(int,int,t2tb2(S),t2tb1(I))) != tb2t1(get(int,int,t2tb2(S),t2tb1(J))) )
& ( \$difference(tb2t1(get(int,int,t2tb2(S),t2tb1(I))),tb2t1(get(int,int,t2tb2(S),t2tb1(J)))) != \$difference(I,J) )
& ( \$difference(tb2t1(get(int,int,t2tb2(S),t2tb1(I))),tb2t1(get(int,int,t2tb2(S),t2tb1(J)))) != \$difference(J,I) ) ) ) ) ) ) ).

tff(partial_solution_eq_prefix,axiom,
! [U: map_int_int,T: map_int_int,K: \$int] :
( partial_solution1(K,T)
=> ( eq_prefix1(int,t2tb2(T),t2tb2(U),K)
=> partial_solution1(K,U) ) ) ).

tff(lt_sol,type,
lt_sol1: ( map_int_int * map_int_int ) > \$o ).

tff(lt_sol_def,axiom,
! [S1: map_int_int,S2: map_int_int] :
( lt_sol1(S1,S2)
<=> ? [I: \$int] :
( \$lesseq(0,I)
& \$less(I,n1)
& eq_prefix1(int,t2tb2(S1),t2tb2(S2),I)
& \$less(tb2t1(get(int,int,t2tb2(S1),t2tb1(I))),tb2t1(get(int,int,t2tb2(S2),t2tb1(I)))) ) ) ).

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

tff(sorted,type,
sorted1: ( map_int_lpmap_int_intrp * \$int * \$int ) > \$o ).

tff(t2tb3,type,
t2tb3: map_int_lpmap_int_intrp > uni ).

tff(t2tb_sort3,axiom,
! [X: map_int_lpmap_int_intrp] : sort1(map(int,map(int,int)),t2tb3(X)) ).

tff(tb2t3,type,
tb2t3: uni > map_int_lpmap_int_intrp ).

tff(bridgeL3,axiom,
! [I: map_int_lpmap_int_intrp] : ( tb2t3(t2tb3(I)) = I ) ).

tff(bridgeR3,axiom,
! [J: uni] : ( t2tb3(tb2t3(J)) = J ) ).

tff(sorted_def,axiom,
! [S: map_int_lpmap_int_intrp,A: \$int,B: \$int] :
( sorted1(S,A,B)
<=> ! [I: \$int,J: \$int] :
( ( \$lesseq(A,I)
& \$less(I,J)
& \$less(J,B) )
=> lt_sol1(tb2t2(get(map(int,int),int,t2tb3(S),t2tb1(I))),tb2t2(get(map(int,int),int,t2tb3(S),t2tb1(J)))) ) ) ).

tff(no_duplicate,axiom,
! [S: map_int_lpmap_int_intrp,A: \$int,B: \$int] :
( sorted1(S,A,B)
=> ! [I: \$int,J: \$int] :
( ( \$lesseq(A,I)
& \$less(I,J)
& \$less(J,B) )
=> ~ eq_prefix1(int,get(map(int,int),int,t2tb3(S),t2tb1(I)),get(map(int,int),int,t2tb3(S),t2tb1(J)),n1) ) ) ).

tff(wP_parameter_queens3,conjecture,
! [Q: \$int,S: \$int,Sol: map_int_lpmap_int_intrp,K: \$int,Col: map_int_int] :
( ( \$lesseq(0,Q)
& ( Q = n1 )
& ( S = 0 )
& ( K = 0 ) )
=> ( ( \$lesseq(0,K)
& ( \$sum(K,cardinal1(int,t2tb(below1(Q)))) = n1 )
& \$lesseq(0,S)
& ! [I: \$int] :
( mem(int,t2tb1(I),t2tb(below1(Q)))
<=> ( \$lesseq(0,I)
& \$less(I,n1)
& ! [J: \$int] :
( ( \$lesseq(0,J)
& \$less(J,K) )
=> ( tb2t1(get(int,int,t2tb2(Col),t2tb1(J))) != I ) ) ) )
& ! [I: \$int] :
( \$lesseq(0,I)
=> ( ~ mem(int,t2tb1(I),empty(int))
<=> ! [J: \$int] :
( ( \$lesseq(0,J)
& \$less(J,K) )
=> ( tb2t1(get(int,int,t2tb2(Col),t2tb1(J))) != \$difference(\$sum(I,J),K) ) ) ) )
& ! [I: \$int] :
( \$lesseq(0,I)
=> ( ~ mem(int,t2tb1(I),empty(int))
<=> ! [J: \$int] :
( ( \$lesseq(0,J)
& \$less(J,K) )
=> ( tb2t1(get(int,int,t2tb2(Col),t2tb1(J))) != \$difference(\$sum(I,K),J) ) ) ) )
& partial_solution1(K,Col) )
=> ! [S1: \$int,Sol1: map_int_lpmap_int_intrp,K1: \$int,Col1: map_int_int] :
( ( \$lesseq(0,\$difference(S1,S))
& ( K1 = K )
& sorted1(Sol1,S,S1)
& ! [T: map_int_int] :
( ( partial_solution1(n1,T)
& eq_prefix1(int,t2tb2(Col1),t2tb2(T),K1) )
<=> ? [I: \$int] :
( \$lesseq(S,I)
& \$less(I,S1)
& eq_prefix1(int,t2tb2(T),get(map(int,int),int,t2tb3(Sol1),t2tb1(I)),n1) ) )
& eq_prefix1(int,t2tb2(Col),t2tb2(Col1),K1)
& eq_prefix1(map(int,int),t2tb3(Sol),t2tb3(Sol1),S) )
=> ( ( \$difference(S1,S) = S1 )
& sorted1(Sol1,0,S1)
& ! [T: map_int_int] :
( partial_solution1(n1,T)
<=> ? [I: \$int] :
( \$lesseq(0,I)
& \$less(I,\$difference(S1,S))
& eq_prefix1(int,t2tb2(T),get(map(int,int),int,t2tb3(Sol1),t2tb1(I)),n1) ) ) ) ) ) ) ).

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