TPTP Problem File: SWW672_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW672_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Vstte12 bfs-T-WP parameter bfs
% 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 : vstte12_bfs-T-WP_parameter_bfs [Fil14]
% Status : Theorem
% Rating : 0.38 v8.2.0, 0.62 v7.5.0, 0.60 v7.4.0, 0.62 v7.3.0, 0.67 v7.0.0, 0.71 v6.4.0, 1.00 v6.3.0, 0.71 v6.2.0, 0.75 v6.1.0
% Syntax : Number of formulae : 102 ( 24 unt; 43 typ; 0 def)
% Number of atoms : 184 ( 38 equ)
% Maximal formula atoms : 26 ( 1 avg)
% Number of connectives : 140 ( 15 ~; 6 |; 30 &)
% ( 14 <=>; 75 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number arithmetic : 56 ( 11 atm; 11 fun; 18 num; 16 var)
% Number of types : 8 ( 6 usr; 1 ari)
% Number of type conns : 65 ( 28 >; 37 *; 0 +; 0 <<)
% Number of predicates : 13 ( 9 usr; 1 prp; 0-6 aty)
% Number of functors : 32 ( 28 usr; 11 con; 0-4 aty)
% Number of variables : 191 ( 184 !; 7 ?; 191 :)
% SPC : TF0_THM_EQU_ARI
% Comments :
%------------------------------------------------------------------------------
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 ) ).
tff(add,type,
add: ( ty * uni * uni ) > uni ).
tff(add_sort1,axiom,
! [A: ty,X: uni,X1: uni] : sort1(set(A),add(A,X,X1)) ).
tff(add_def1,axiom,
! [A: ty,X: uni,Y: uni] :
( sort1(A,X)
=> ( sort1(A,Y)
=> ! [S: uni] :
( mem(A,X,add(A,Y,S))
<=> ( ( 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) ) ) ) ) ).
tff(add_remove,axiom,
! [A: ty,X: uni,S: uni] :
( sort1(set(A),S)
=> ( mem(A,X,S)
=> ( add(A,X,remove(A,X,S)) = S ) ) ) ).
tff(remove_add,axiom,
! [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) ) ).
tff(cardinal_add,axiom,
! [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(vertex,type,
vertex1: $tType ).
tff(vertex1,type,
vertex: ty ).
tff(set_vertex,type,
set_vertex: $tType ).
tff(succ,type,
succ1: vertex1 > set_vertex ).
tff(path,type,
path1: ( vertex1 * vertex1 * $int ) > $o ).
tff(path_empty,axiom,
! [V: vertex1] : path1(V,V,0) ).
tff(t2tb,type,
t2tb: set_vertex > uni ).
tff(t2tb_sort,axiom,
! [X: set_vertex] : sort1(set(vertex),t2tb(X)) ).
tff(tb2t,type,
tb2t: uni > set_vertex ).
tff(bridgeL,axiom,
! [I: set_vertex] : tb2t(t2tb(I)) = I ).
tff(bridgeR,axiom,
! [J: uni] :
( sort1(set(vertex),J)
=> ( t2tb(tb2t(J)) = J ) ) ).
tff(t2tb1,type,
t2tb1: vertex1 > uni ).
tff(t2tb_sort1,axiom,
! [X: vertex1] : sort1(vertex,t2tb1(X)) ).
tff(tb2t1,type,
tb2t1: uni > vertex1 ).
tff(bridgeL1,axiom,
! [I: vertex1] : tb2t1(t2tb1(I)) = I ).
tff(bridgeR1,axiom,
! [J: uni] :
( sort1(vertex,J)
=> ( t2tb1(tb2t1(J)) = J ) ) ).
tff(path_succ,axiom,
! [V1: vertex1,V2: vertex1,V3: vertex1,N: $int] :
( path1(V1,V2,N)
=> ( mem(vertex,t2tb1(V3),t2tb(succ1(V2)))
=> path1(V1,V3,$sum(N,1)) ) ) ).
tff(path_inversion,axiom,
! [Z: vertex1,Z1: vertex1,Z2: $int] :
( path1(Z,Z1,Z2)
=> ( ? [V: vertex1] :
( ( Z = V )
& ( Z1 = V )
& ( Z2 = 0 ) )
| ? [V1: vertex1,V2: vertex1,V3: vertex1,N: $int] :
( path1(V1,V2,N)
& mem(vertex,t2tb1(V3),t2tb(succ1(V2)))
& ( Z = V1 )
& ( Z1 = V3 )
& ( Z2 = $sum(N,1) ) ) ) ) ).
tff(path_nonneg,axiom,
! [V1: vertex1,V2: vertex1,N: $int] :
( path1(V1,V2,N)
=> $lesseq(0,N) ) ).
tff(path_inversion1,axiom,
! [V1: vertex1,V3: vertex1,N: $int] :
( $lesseq(0,N)
=> ( path1(V1,V3,$sum(N,1))
=> ? [V2: vertex1] :
( path1(V1,V2,N)
& mem(vertex,t2tb1(V3),t2tb(succ1(V2))) ) ) ) ).
tff(path_closure,axiom,
! [S: set_vertex] :
( ! [X: vertex1] :
( mem(vertex,t2tb1(X),t2tb(S))
=> ! [Y: vertex1] :
( mem(vertex,t2tb1(Y),t2tb(succ1(X)))
=> mem(vertex,t2tb1(Y),t2tb(S)) ) )
=> ! [V1: vertex1,V2: vertex1,N: $int] :
( path1(V1,V2,N)
=> ( mem(vertex,t2tb1(V1),t2tb(S))
=> mem(vertex,t2tb1(V2),t2tb(S)) ) ) ) ).
tff(shortest_path,type,
shortest_path1: ( vertex1 * vertex1 * $int ) > $o ).
tff(shortest_path_def,axiom,
! [V1: vertex1,V2: vertex1,N: $int] :
( shortest_path1(V1,V2,N)
<=> ( path1(V1,V2,N)
& ! [M: $int] :
( $less(M,N)
=> ~ path1(V1,V2,M) ) ) ) ).
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(inv,type,
inv1: ( vertex1 * vertex1 * set_vertex * set_vertex * set_vertex * $int ) > $o ).
tff(inv_def,axiom,
! [S: vertex1,T: vertex1,Visited: set_vertex,Current: set_vertex,Next: set_vertex,D: $int] :
( inv1(S,T,Visited,Current,Next,D)
<=> ( subset(vertex,t2tb(Current),t2tb(Visited))
& ! [X: vertex1] :
( mem(vertex,t2tb1(X),t2tb(Current))
=> shortest_path1(S,X,D) )
& subset(vertex,t2tb(Next),t2tb(Visited))
& ! [X: vertex1] :
( mem(vertex,t2tb1(X),t2tb(Next))
=> shortest_path1(S,X,$sum(D,1)) )
& ! [X: vertex1,M: $int] :
( path1(S,X,M)
=> ( $lesseq(M,D)
=> mem(vertex,t2tb1(X),t2tb(Visited)) ) )
& ! [X: vertex1] :
( mem(vertex,t2tb1(X),t2tb(Visited))
=> ? [M: $int] :
( path1(S,X,M)
& $lesseq(M,$sum(D,1)) ) )
& ! [X: vertex1] :
( shortest_path1(S,X,$sum(D,1))
=> ( mem(vertex,t2tb1(X),t2tb(Next))
| ~ mem(vertex,t2tb1(X),t2tb(Visited)) ) )
& ( mem(vertex,t2tb1(T),t2tb(Visited))
=> ( mem(vertex,t2tb1(T),t2tb(Current))
| mem(vertex,t2tb1(T),t2tb(Next)) ) ) ) ) ).
tff(closure,type,
closure1: ( set_vertex * set_vertex * set_vertex * vertex1 ) > $o ).
tff(closure_def,axiom,
! [Visited: set_vertex,Current: set_vertex,Next: set_vertex,X: vertex1] :
( closure1(Visited,Current,Next,X)
<=> ( mem(vertex,t2tb1(X),t2tb(Visited))
=> ( ~ mem(vertex,t2tb1(X),t2tb(Current))
=> ( ~ mem(vertex,t2tb1(X),t2tb(Next))
=> ! [Y: vertex1] :
( mem(vertex,t2tb1(Y),t2tb(succ1(X)))
=> mem(vertex,t2tb1(Y),t2tb(Visited)) ) ) ) ) ) ).
tff(wP_parameter_bfs,conjecture,
! [S: vertex1,T: vertex1,D: $int,Next: set_vertex,Current: set_vertex,Visited: set_vertex] :
( ( inv1(S,T,Visited,Current,Next,D)
& ( is_empty(vertex,t2tb(Current))
=> is_empty(vertex,t2tb(Next)) )
& ! [X: vertex1] : closure1(Visited,Current,Next,X)
& $lesseq(0,D) )
=> ! [O: bool1] :
( ( ( O = true1 )
<=> is_empty(vertex,t2tb(Current)) )
=> ( ( O != true1 )
=> ( ~ is_empty(vertex,t2tb(Current))
=> ! [Current1: set_vertex,V: vertex1] :
( ( mem(vertex,t2tb1(V),t2tb(Current))
& ( Current1 = tb2t(remove(vertex,t2tb1(V),t2tb(Current))) ) )
=> ( ( V != T )
=> ( ( inv1(S,T,Visited,Current1,Next,D)
& shortest_path1(S,V,D)
& ! [X: vertex1] :
( ( X != V )
=> closure1(Visited,Current1,Next,X) ) )
=> ! [Next1: set_vertex,Visited1: set_vertex] :
( ( inv1(S,T,Visited1,Current1,Next1,D)
& subset(vertex,t2tb(succ1(V)),t2tb(Visited1))
& ! [X: vertex1] : closure1(Visited1,Current1,Next1,X) )
=> ! [Result: bool1] :
( ( ( Result = true1 )
<=> is_empty(vertex,t2tb(Current1)) )
=> ( ( Result = true1 )
=> ! [Current2: set_vertex] :
( ( Current2 = Next1 )
=> ! [Next2: set_vertex] :
( ( Next2 = tb2t(empty(vertex)) )
=> ! [D1: $int] :
( ( D1 = $sum(D,1) )
=> ! [X: vertex1] : closure1(Visited1,Current2,Next2,X) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------