TPTP Problem File: SWW587_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW587_2 : TPTP v8.2.0. Released v6.1.0.
% Domain : Software Verification
% Problem : Dijkstra-T-WP parameter shortest path code
% 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 : dijkstra-T-WP_parameter_shortest_path_code [Fil14]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.62 v7.5.0, 0.60 v7.4.0, 0.62 v7.3.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.33 v6.3.0, 0.57 v6.2.0, 1.00 v6.1.0
% Syntax : Number of formulae : 135 ( 35 unt; 57 typ; 0 def)
% Number of atoms : 288 ( 67 equ)
% Maximal formula atoms : 90 ( 2 avg)
% Number of connectives : 236 ( 26 ~; 19 |; 68 &)
% ( 17 <=>; 106 =>; 0 <=; 0 <~>)
% Maximal formula depth : 46 ( 6 avg)
% Maximal term depth : 8 ( 1 avg)
% Number arithmetic : 76 ( 23 atm; 18 fun; 16 num; 19 var)
% Number of types : 9 ( 7 usr; 1 ari)
% Number of type conns : 95 ( 40 >; 55 *; 0 +; 0 <<)
% Number of predicates : 16 ( 12 usr; 1 prp; 0-6 aty)
% Number of functors : 43 ( 38 usr; 12 con; 0-5 aty)
% Number of variables : 273 ( 264 !; 9 ?; 273 :)
% 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(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(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(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(vertex,type,
vertex1: $tType ).
tff(vertex1,type,
vertex: ty ).
tff(set_vertex,type,
set_vertex: $tType ).
tff(v,type,
v1: set_vertex ).
tff(g_succ,type,
g_succ1: vertex1 > set_vertex ).
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(g_succ_sound,axiom,
! [X: vertex1] : subset(vertex,t2tb(g_succ1(X)),t2tb(v1)) ).
tff(weight,type,
weight1: ( vertex1 * vertex1 ) > $int ).
tff(weight_nonneg,axiom,
! [X: vertex1,Y: vertex1] : $lesseq(0,weight1(X,Y)) ).
tff(map_vertex_int,type,
map_vertex_int: $tType ).
tff(min,type,
min1: ( vertex1 * set_vertex * map_vertex_int ) > $o ).
tff(t2tb1,type,
t2tb1: map_vertex_int > uni ).
tff(t2tb_sort1,axiom,
! [X: map_vertex_int] : sort1(map(vertex,int),t2tb1(X)) ).
tff(tb2t1,type,
tb2t1: uni > map_vertex_int ).
tff(bridgeL1,axiom,
! [I: map_vertex_int] : tb2t1(t2tb1(I)) = I ).
tff(bridgeR1,axiom,
! [J: uni] : t2tb1(tb2t1(J)) = J ).
tff(t2tb2,type,
t2tb2: vertex1 > uni ).
tff(t2tb_sort2,axiom,
! [X: vertex1] : sort1(vertex,t2tb2(X)) ).
tff(tb2t2,type,
tb2t2: uni > vertex1 ).
tff(bridgeL2,axiom,
! [I: vertex1] : tb2t2(t2tb2(I)) = I ).
tff(bridgeR2,axiom,
! [J: uni] :
( sort1(vertex,J)
=> ( t2tb2(tb2t2(J)) = J ) ) ).
tff(t2tb3,type,
t2tb3: $int > uni ).
tff(t2tb_sort3,axiom,
! [X: $int] : sort1(int,t2tb3(X)) ).
tff(tb2t3,type,
tb2t3: uni > $int ).
tff(bridgeL3,axiom,
! [I: $int] : tb2t3(t2tb3(I)) = I ).
tff(bridgeR3,axiom,
! [J: uni] : t2tb3(tb2t3(J)) = J ).
tff(min_def,axiom,
! [M: vertex1,Q: set_vertex,D: map_vertex_int] :
( min1(M,Q,D)
<=> ( mem(vertex,t2tb2(M),t2tb(Q))
& ! [X: vertex1] :
( mem(vertex,t2tb2(X),t2tb(Q))
=> $lesseq(tb2t3(get(int,vertex,t2tb1(D),t2tb2(M))),tb2t3(get(int,vertex,t2tb1(D),t2tb2(X)))) ) ) ) ).
tff(path,type,
path1: ( vertex1 * vertex1 * $int ) > $o ).
tff(path_nil,axiom,
! [X: vertex1] : path1(X,X,0) ).
tff(path_cons,axiom,
! [X: vertex1,Y: vertex1,Z: vertex1,D: $int] :
( path1(X,Y,D)
=> ( mem(vertex,t2tb2(Z),t2tb(g_succ1(Y)))
=> path1(X,Z,$sum(D,weight1(Y,Z))) ) ) ).
tff(path_inversion,axiom,
! [Z: vertex1,Z1: vertex1,Z2: $int] :
( path1(Z,Z1,Z2)
=> ( ? [X: vertex1] :
( ( Z = X )
& ( Z1 = X )
& ( Z2 = 0 ) )
| ? [X: vertex1,Y: vertex1,Z3: vertex1,D: $int] :
( path1(X,Y,D)
& mem(vertex,t2tb2(Z3),t2tb(g_succ1(Y)))
& ( Z = X )
& ( Z1 = Z3 )
& ( Z2 = $sum(D,weight1(Y,Z3)) ) ) ) ) ).
tff(length_nonneg,axiom,
! [X: vertex1,Y: vertex1,D: $int] :
( path1(X,Y,D)
=> $lesseq(0,D) ) ).
tff(shortest_path,type,
shortest_path1: ( vertex1 * vertex1 * $int ) > $o ).
tff(shortest_path_def,axiom,
! [X: vertex1,Y: vertex1,D: $int] :
( shortest_path1(X,Y,D)
<=> ( path1(X,Y,D)
& ! [Dqt: $int] :
( path1(X,Y,Dqt)
=> $lesseq(D,Dqt) ) ) ) ).
tff(path_inversion1,axiom,
! [Src: vertex1,V: vertex1,D: $int] :
( path1(Src,V,D)
=> ( ( ( V = Src )
& ( D = 0 ) )
| ? [Vqt: vertex1] :
( path1(Src,Vqt,$difference(D,weight1(Vqt,V)))
& mem(vertex,t2tb2(V),t2tb(g_succ1(Vqt))) ) ) ) ).
tff(path_shortest_path,axiom,
! [Src: vertex1,V: vertex1,D: $int] :
( path1(Src,V,D)
=> ? [Dqt: $int] :
( shortest_path1(Src,V,Dqt)
& $lesseq(Dqt,D) ) ) ).
tff(main_lemma,axiom,
! [Src: vertex1,V: vertex1,D: $int] :
( path1(Src,V,D)
=> ( ~ shortest_path1(Src,V,D)
=> ( ( ( V = Src )
& $less(0,D) )
| ? [Vqt: vertex1,Dqt: $int] :
( shortest_path1(Src,Vqt,Dqt)
& mem(vertex,t2tb2(V),t2tb(g_succ1(Vqt)))
& $less($sum(Dqt,weight1(Vqt,V)),D) ) ) ) ) ).
tff(completeness_lemma,axiom,
! [S: set_vertex] :
( ! [V: vertex1] :
( mem(vertex,t2tb2(V),t2tb(S))
=> ! [W: vertex1] :
( mem(vertex,t2tb2(W),t2tb(g_succ1(V)))
=> mem(vertex,t2tb2(W),t2tb(S)) ) )
=> ! [Src: vertex1] :
( mem(vertex,t2tb2(Src),t2tb(S))
=> ! [Dst: vertex1,D: $int] :
( path1(Src,Dst,D)
=> mem(vertex,t2tb2(Dst),t2tb(S)) ) ) ) ).
tff(inv_src,type,
inv_src1: ( vertex1 * set_vertex * set_vertex ) > $o ).
tff(inv_src_def,axiom,
! [Src: vertex1,S: set_vertex,Q: set_vertex] :
( inv_src1(Src,S,Q)
<=> ( mem(vertex,t2tb2(Src),t2tb(S))
| mem(vertex,t2tb2(Src),t2tb(Q)) ) ) ).
tff(inv,type,
inv1: ( vertex1 * set_vertex * set_vertex * map_vertex_int ) > $o ).
tff(inv_def,axiom,
! [Src: vertex1,S: set_vertex,Q: set_vertex,D: map_vertex_int] :
( inv1(Src,S,Q,D)
<=> ( inv_src1(Src,S,Q)
& ( tb2t3(get(int,vertex,t2tb1(D),t2tb2(Src))) = 0 )
& subset(vertex,t2tb(S),t2tb(v1))
& subset(vertex,t2tb(Q),t2tb(v1))
& ! [V: vertex1] :
( mem(vertex,t2tb2(V),t2tb(Q))
=> ( mem(vertex,t2tb2(V),t2tb(S))
=> $false ) )
& ! [V: vertex1] :
( mem(vertex,t2tb2(V),t2tb(S))
=> shortest_path1(Src,V,tb2t3(get(int,vertex,t2tb1(D),t2tb2(V)))) )
& ! [V: vertex1] :
( mem(vertex,t2tb2(V),t2tb(Q))
=> path1(Src,V,tb2t3(get(int,vertex,t2tb1(D),t2tb2(V)))) ) ) ) ).
tff(inv_succ,type,
inv_succ1: ( vertex1 * set_vertex * set_vertex * map_vertex_int ) > $o ).
tff(inv_succ_def,axiom,
! [Src: vertex1,S: set_vertex,Q: set_vertex,D: map_vertex_int] :
( inv_succ1(Src,S,Q,D)
<=> ! [X: vertex1] :
( mem(vertex,t2tb2(X),t2tb(S))
=> ! [Y: vertex1] :
( mem(vertex,t2tb2(Y),t2tb(g_succ1(X)))
=> ( ( mem(vertex,t2tb2(Y),t2tb(S))
| mem(vertex,t2tb2(Y),t2tb(Q)) )
& $lesseq(tb2t3(get(int,vertex,t2tb1(D),t2tb2(Y))),$sum(tb2t3(get(int,vertex,t2tb1(D),t2tb2(X))),weight1(X,Y))) ) ) ) ) ).
tff(inv_succ2,type,
inv_succ21: ( vertex1 * set_vertex * set_vertex * map_vertex_int * vertex1 * set_vertex ) > $o ).
tff(inv_succ2_def,axiom,
! [Src: vertex1,S: set_vertex,Q: set_vertex,D: map_vertex_int,U: vertex1,Su: set_vertex] :
( inv_succ21(Src,S,Q,D,U,Su)
<=> ! [X: vertex1] :
( mem(vertex,t2tb2(X),t2tb(S))
=> ! [Y: vertex1] :
( mem(vertex,t2tb2(Y),t2tb(g_succ1(X)))
=> ( ( ( X != U )
| ( ( X = U )
& ~ mem(vertex,t2tb2(Y),t2tb(Su)) ) )
=> ( ( mem(vertex,t2tb2(Y),t2tb(S))
| mem(vertex,t2tb2(Y),t2tb(Q)) )
& $lesseq(tb2t3(get(int,vertex,t2tb1(D),t2tb2(Y))),$sum(tb2t3(get(int,vertex,t2tb1(D),t2tb2(X))),weight1(X,Y))) ) ) ) ) ) ).
tff(wP_parameter_shortest_path_code,conjecture,
! [Src: vertex1,Dst: vertex1,D: map_vertex_int] :
( ( mem(vertex,t2tb2(Src),t2tb(v1))
& mem(vertex,t2tb2(Dst),t2tb(v1)) )
=> ! [Q: set_vertex,D1: map_vertex_int,Visited: set_vertex] :
( ( ! [X: vertex1] : ~ mem(vertex,t2tb2(X),t2tb(Visited))
& ( Q = tb2t(add(vertex,t2tb2(Src),empty(vertex))) )
& ( D1 = tb2t1(set1(int,vertex,t2tb1(D),t2tb2(Src),t2tb3(0))) ) )
=> ! [Q1: set_vertex,D2: map_vertex_int,Visited1: set_vertex] :
( ( inv_src1(Src,Visited1,Q1)
& ( tb2t3(get(int,vertex,t2tb1(D2),t2tb2(Src))) = 0 )
& subset(vertex,t2tb(Visited1),t2tb(v1))
& subset(vertex,t2tb(Q1),t2tb(v1))
& ! [V: vertex1] :
( mem(vertex,t2tb2(V),t2tb(Q1))
=> ( mem(vertex,t2tb2(V),t2tb(Visited1))
=> $false ) )
& ! [V: vertex1] :
( mem(vertex,t2tb2(V),t2tb(Visited1))
=> shortest_path1(Src,V,tb2t3(get(int,vertex,t2tb1(D2),t2tb2(V)))) )
& ! [V: vertex1] :
( mem(vertex,t2tb2(V),t2tb(Q1))
=> path1(Src,V,tb2t3(get(int,vertex,t2tb1(D2),t2tb2(V)))) )
& ! [X: vertex1] :
( mem(vertex,t2tb2(X),t2tb(Visited1))
=> ! [Y: vertex1] :
( mem(vertex,t2tb2(Y),t2tb(g_succ1(X)))
=> ( ( mem(vertex,t2tb2(Y),t2tb(Visited1))
| mem(vertex,t2tb2(Y),t2tb(Q1)) )
& $lesseq(tb2t3(get(int,vertex,t2tb1(D2),t2tb2(Y))),$sum(tb2t3(get(int,vertex,t2tb1(D2),t2tb2(X))),weight1(X,Y))) ) ) )
& ! [M: vertex1] :
( ( mem(vertex,t2tb2(M),t2tb(Q1))
& ! [X: vertex1] :
( mem(vertex,t2tb2(X),t2tb(Q1))
=> $lesseq(tb2t3(get(int,vertex,t2tb1(D2),t2tb2(M))),tb2t3(get(int,vertex,t2tb1(D2),t2tb2(X)))) ) )
=> ! [X: vertex1,Dx: $int] :
( path1(Src,X,Dx)
=> ( $less(Dx,tb2t3(get(int,vertex,t2tb1(D2),t2tb2(M))))
=> mem(vertex,t2tb2(X),t2tb(Visited1)) ) ) ) )
=> ! [O: bool1] :
( ( ( O = true1 )
<=> ! [X: vertex1] : ~ mem(vertex,t2tb2(X),t2tb(Q1)) )
=> ( ( O != true1 )
=> ( ~ ! [X: vertex1] : ~ mem(vertex,t2tb2(X),t2tb(Q1))
=> ! [Q2: set_vertex,U: vertex1] :
( ( mem(vertex,t2tb2(U),t2tb(Q1))
& ! [X: vertex1] :
( mem(vertex,t2tb2(X),t2tb(Q1))
=> $lesseq(tb2t3(get(int,vertex,t2tb1(D2),t2tb2(U))),tb2t3(get(int,vertex,t2tb1(D2),t2tb2(X)))) )
& ( Q2 = tb2t(remove(vertex,t2tb2(U),t2tb(Q1))) ) )
=> ( ( path1(Src,U,tb2t3(get(int,vertex,t2tb1(D2),t2tb2(U))))
& ! [Dqt: $int] :
( path1(Src,U,Dqt)
=> $lesseq(tb2t3(get(int,vertex,t2tb1(D2),t2tb2(U))),Dqt) ) )
=> ! [Visited2: set_vertex] :
( ( Visited2 = tb2t(add(vertex,t2tb2(U),t2tb(Visited1))) )
=> ! [Su: set_vertex,Q3: set_vertex,D3: map_vertex_int] :
( ( ! [X: vertex1] :
( mem(vertex,t2tb2(X),t2tb(Su))
=> mem(vertex,t2tb2(X),t2tb(g_succ1(U))) )
& inv_src1(Src,Visited2,Q3)
& ( tb2t3(get(int,vertex,t2tb1(D3),t2tb2(Src))) = 0 )
& subset(vertex,t2tb(Visited2),t2tb(v1))
& subset(vertex,t2tb(Q3),t2tb(v1))
& ! [V: vertex1] :
( mem(vertex,t2tb2(V),t2tb(Q3))
=> ( mem(vertex,t2tb2(V),t2tb(Visited2))
=> $false ) )
& ! [V: vertex1] :
( mem(vertex,t2tb2(V),t2tb(Visited2))
=> shortest_path1(Src,V,tb2t3(get(int,vertex,t2tb1(D3),t2tb2(V)))) )
& ! [V: vertex1] :
( mem(vertex,t2tb2(V),t2tb(Q3))
=> path1(Src,V,tb2t3(get(int,vertex,t2tb1(D3),t2tb2(V)))) )
& ! [X: vertex1] :
( mem(vertex,t2tb2(X),t2tb(Visited2))
=> ! [Y: vertex1] :
( mem(vertex,t2tb2(Y),t2tb(g_succ1(X)))
=> ( ( ( X != U )
| ( ( X = U )
& ~ mem(vertex,t2tb2(Y),t2tb(Su)) ) )
=> ( ( mem(vertex,t2tb2(Y),t2tb(Visited2))
| mem(vertex,t2tb2(Y),t2tb(Q3)) )
& $lesseq(tb2t3(get(int,vertex,t2tb1(D3),t2tb2(Y))),$sum(tb2t3(get(int,vertex,t2tb1(D3),t2tb2(X))),weight1(X,Y))) ) ) ) ) )
=> ! [Result: bool1] :
( ( ( Result = true1 )
<=> ~ ! [X: vertex1] : ~ mem(vertex,t2tb2(X),t2tb(Su)) )
=> ( ( Result = true1 )
=> ( ~ ! [X: vertex1] : ~ mem(vertex,t2tb2(X),t2tb(Su))
=> ! [Su1: set_vertex,V: vertex1] :
( ( mem(vertex,t2tb2(V),t2tb(Su))
& ( Su1 = tb2t(remove(vertex,t2tb2(V),t2tb(Su))) ) )
=> ! [Q4: set_vertex,D4: map_vertex_int] :
( ( ( mem(vertex,t2tb2(V),t2tb(Visited2))
& ( Q4 = Q3 )
& ( D4 = D3 ) )
| ( mem(vertex,t2tb2(V),t2tb(Q4))
& $lesseq(tb2t3(get(int,vertex,t2tb1(D4),t2tb2(V))),$sum(tb2t3(get(int,vertex,t2tb1(D4),t2tb2(U))),weight1(U,V)))
& ( Q4 = Q3 )
& ( D4 = D3 ) )
| ( mem(vertex,t2tb2(V),t2tb(Q4))
& $less($sum(tb2t3(get(int,vertex,t2tb1(D3),t2tb2(U))),weight1(U,V)),tb2t3(get(int,vertex,t2tb1(D3),t2tb2(V))))
& ( Q4 = Q3 )
& ( D4 = tb2t1(set1(int,vertex,t2tb1(D3),t2tb2(V),t2tb3($sum(tb2t3(get(int,vertex,t2tb1(D3),t2tb2(U))),weight1(U,V))))) ) )
| ( ~ mem(vertex,t2tb2(V),t2tb(Visited2))
& ~ mem(vertex,t2tb2(V),t2tb(Q3))
& ( Q4 = tb2t(add(vertex,t2tb2(V),t2tb(Q3))) )
& ( D4 = tb2t1(set1(int,vertex,t2tb1(D3),t2tb2(V),t2tb3($sum(tb2t3(get(int,vertex,t2tb1(D3),t2tb2(U))),weight1(U,V))))) ) ) )
=> ( ( $less(tb2t3(get(int,vertex,t2tb1(D4),t2tb2(V))),$sum(tb2t3(get(int,vertex,t2tb1(D4),t2tb2(U))),weight1(U,V)))
| ( tb2t3(get(int,vertex,t2tb1(D4),t2tb2(V))) = $sum(tb2t3(get(int,vertex,t2tb1(D4),t2tb2(U))),weight1(U,V)) ) )
=> ! [X: vertex1] :
( mem(vertex,t2tb2(X),t2tb(Visited2))
=> ! [Y: vertex1] :
( mem(vertex,t2tb2(Y),t2tb(g_succ1(X)))
=> ( ( ( X != U )
| ( ( X = U )
& ~ mem(vertex,t2tb2(Y),t2tb(Su1)) ) )
=> ( mem(vertex,t2tb2(Y),t2tb(Visited2))
| mem(vertex,t2tb2(Y),t2tb(Q4)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------