## TPTP Problem File: SWW637=2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SWW637=2 : TPTP v8.0.0. Released v6.1.0.
% Domain   : Software Verification
% Problem  : Relabel-T-WP parameter relabel
% 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    : relabel-T-WP_parameter_relabel [Fil14]

% Status   : Theorem
% Rating   : 0.75 v7.5.0, 0.80 v7.4.0, 0.75 v7.3.0, 0.67 v7.0.0, 0.71 v6.4.0, 1.00 v6.3.0, 0.86 v6.2.0, 1.00 v6.1.0
% Syntax   : Number of formulae    :  122 (  43 unt;  50 typ;   0 def)
%            Number of atoms       :  152 (  52 equ)
%            Maximal formula atoms :   18 (   1 avg)
%            Number of connectives :   87 (   7   ~;   9   |;  33   &)
%                                         (   5 <=>;  33  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   5 avg)
%            Maximal term depth    :    4 (   1 avg)
%            Number arithmetic     :   33 (  13 atm;   4 fun;   5 num;  11 var)
%            Number of types       :   10 (   8 usr;   1 ari)
%            Number of type conns  :   67 (  33   >;  34   *;   0   +;   0  <<)
%            Number of predicates  :    7 (   4 usr;   0 prp; 2-4 aty)
%            Number of functors    :   42 (  38 usr;  11 con; 0-5 aty)
%            Number of variables   :  205 ( 194   !;  11   ?; 205   :)
% 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(list,type,
list: ty > ty ).

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

tff(nil_sort,axiom,
! [A: ty] : sort(list(A),nil(A)) ).

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

tff(cons_sort,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_sort,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_sort,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_sort,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(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(infix_plpl,type,
infix_plpl: ( ty * uni * uni ) > uni ).

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

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(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_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(distinct,type,
distinct: ( ty * uni ) > \$o ).

tff(distinct_zero,axiom,
! [A: ty] : distinct(A,nil(A)) ).

tff(distinct_one,axiom,
! [A: ty,X: uni] : distinct(A,cons(A,X,nil(A))) ).

tff(distinct_many,axiom,
! [A: ty,X: uni,L: uni] :
( ~ mem(A,X,L)
=> ( distinct(A,L)
=> distinct(A,cons(A,X,L)) ) ) ).

tff(distinct_inversion,axiom,
! [A: ty,Z: uni] :
( distinct(A,Z)
=> ( ( Z = nil(A) )
| ? [X: uni] :
( sort(A,X)
& ( Z = cons(A,X,nil(A)) ) )
| ? [X: uni,L: uni] :
( sort(A,X)
& sort(list(A),L)
& ~ mem(A,X,L)
& distinct(A,L)
& ( Z = cons(A,X,L) ) ) ) ) ).

tff(distinct_append,axiom,
! [A: ty,L1: uni,L2: uni] :
( distinct(A,L1)
=> ( distinct(A,L2)
=> ( ! [X: uni] :
( sort(A,X)
=> ( mem(A,X,L1)
=> ~ mem(A,X,L2) ) )
=> distinct(A,infix_plpl(A,L1,L2)) ) ) ) ).

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

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

tff(leaf_sort,axiom,
! [A: ty,X: uni] : sort(tree(A),leaf(A,X)) ).

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

tff(node_sort,axiom,
! [A: ty,X: uni,X1: uni] : sort(tree(A),node(A,X,X1)) ).

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

tff(match_tree_sort,axiom,
! [A: ty,A1: ty,X: uni,X1: uni,X2: uni] : sort(A1,match_tree(A1,A,X,X1,X2)) ).

tff(match_tree_Leaf,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni,U: uni] :
( sort(A1,Z)
=> ( match_tree(A1,A,leaf(A,U),Z,Z1) = Z ) ) ).

tff(match_tree_Node,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni,U: uni,U1: uni] :
( sort(A1,Z1)
=> ( match_tree(A1,A,node(A,U,U1),Z,Z1) = Z1 ) ) ).

tff(leaf_Node,axiom,
! [A: ty,U: uni,V: uni,V1: uni] : ( leaf(A,U) != node(A,V,V1) ) ).

tff(leaf_proj_1,type,
leaf_proj_1: ( ty * uni ) > uni ).

tff(leaf_proj_1_sort,axiom,
! [A: ty,X: uni] : sort(A,leaf_proj_1(A,X)) ).

tff(leaf_proj_1_def,axiom,
! [A: ty,U: uni] :
( sort(A,U)
=> ( leaf_proj_1(A,leaf(A,U)) = U ) ) ).

tff(node_proj_1,type,
node_proj_1: ( ty * uni ) > uni ).

tff(node_proj_1_sort,axiom,
! [A: ty,X: uni] : sort(tree(A),node_proj_1(A,X)) ).

tff(node_proj_1_def,axiom,
! [A: ty,U: uni,U1: uni] : ( node_proj_1(A,node(A,U,U1)) = U ) ).

tff(node_proj_2,type,
node_proj_2: ( ty * uni ) > uni ).

tff(node_proj_2_sort,axiom,
! [A: ty,X: uni] : sort(tree(A),node_proj_2(A,X)) ).

tff(node_proj_2_def,axiom,
! [A: ty,U: uni,U1: uni] : ( node_proj_2(A,node(A,U,U1)) = U1 ) ).

tff(tree_inversion,axiom,
! [A: ty,U: uni] :
( ( U = leaf(A,leaf_proj_1(A,U)) )
| ( U = node(A,node_proj_1(A,U),node_proj_2(A,U)) ) ) ).

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

tff(labels_sort,axiom,
! [A: ty,X: uni] : sort(list(A),labels(A,X)) ).

tff(labels_def,axiom,
! [A: ty] :
( ! [X: uni] : ( labels(A,leaf(A,X)) = cons(A,X,nil(A)) )
& ! [X: uni,X1: uni] : ( labels(A,node(A,X,X1)) = infix_plpl(A,labels(A,X),labels(A,X1)) ) ) ).

tff(labels_Leaf,axiom,
! [A: ty,X: uni,Y: uni] :
( sort(A,X)
=> ( sort(A,Y)
=> ( mem(A,X,labels(A,leaf(A,Y)))
<=> ( X = Y ) ) ) ) ).

tff(labels_Node,axiom,
! [A: ty,X: uni,L: uni,R: uni] :
( mem(A,X,labels(A,node(A,L,R)))
<=> ( mem(A,X,labels(A,L))
| mem(A,X,labels(A,R)) ) ) ).

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

tff(same_shape_Leaf,axiom,
! [A: ty,B: ty,X1: uni,X2: uni] : same_shape(B,A,leaf(A,X1),leaf(B,X2)) ).

tff(same_shape_Node,axiom,
! [A: ty,B: ty,L1: uni,R1: uni,L2: uni,R2: uni] :
( same_shape(B,A,L1,L2)
=> ( same_shape(B,A,R1,R2)
=> same_shape(B,A,node(A,L1,R1),node(B,L2,R2)) ) ) ).

tff(same_shape_inversion,axiom,
! [A: ty,B: ty,Z: uni,Z1: uni] :
( same_shape(B,A,Z,Z1)
=> ( ? [X1: uni,X2: uni] :
( sort(A,X1)
& sort(B,X2)
& ( Z = leaf(A,X1) )
& ( Z1 = leaf(B,X2) ) )
| ? [L1: uni,R1: uni,L2: uni,R2: uni] :
( sort(tree(A),L1)
& sort(tree(A),R1)
& sort(tree(B),L2)
& sort(tree(B),R2)
& same_shape(B,A,L1,L2)
& same_shape(B,A,R1,R2)
& ( Z = node(A,L1,R1) )
& ( Z1 = node(B,L2,R2) ) ) ) ) ).

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(a,type,
a: \$tType ).

tff(a1,type,
a1: ty ).

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

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

tff(t2tb_sort,axiom,
! [X: tree_int] : sort(tree(int),t2tb(X)) ).

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

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

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

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

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

tff(t2tb_sort1,axiom,
! [X: list_int] : sort(list(int),t2tb1(X)) ).

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

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

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

tff(tree_a1,type,
tree_a1: \$tType ).

tff(t2tb2,type,
t2tb2: tree_a1 > uni ).

tff(t2tb_sort2,axiom,
! [X: tree_a1] : sort(tree(a1),t2tb2(X)) ).

tff(tb2t2,type,
tb2t2: uni > tree_a1 ).

tff(bridgeL2,axiom,
! [I: tree_a1] : ( tb2t2(t2tb2(I)) = I ) ).

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

tff(t2tb3,type,
t2tb3: \$int > uni ).

tff(t2tb_sort3,axiom,
! [X: \$int] : sort(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(wP_parameter_relabel,conjecture,
! [R: \$int,X: tree_a1,X1: tree_a1,R1: \$int,O: tree_int] :
( ( same_shape(int,a1,t2tb2(X1),t2tb(O))
& distinct(int,labels(int,t2tb(O)))
& \$lesseq(R,R1)
& ! [X2: \$int] :
( mem(int,t2tb3(X2),labels(int,t2tb(O)))
=> ( \$less(R,X2)
& \$lesseq(X2,R1) ) ) )
=> ! [R2: \$int,O1: tree_int] :
( ( same_shape(int,a1,t2tb2(X),t2tb(O1))
& distinct(int,labels(int,t2tb(O1)))
& \$lesseq(R1,R2)
& ! [X2: \$int] :
( mem(int,t2tb3(X2),labels(int,t2tb(O1)))
=> ( \$less(R1,X2)
& \$lesseq(X2,R2) ) ) )
=> ( same_shape(int,a1,node(a1,t2tb2(X),t2tb2(X1)),node(int,t2tb(O1),t2tb(O)))
& distinct(int,labels(int,node(int,t2tb(O1),t2tb(O))))
& \$lesseq(R,R2)
& ! [X2: \$int] :
( mem(int,t2tb3(X2),labels(int,node(int,t2tb(O1),t2tb(O))))
=> ( \$less(R,X2)
& \$lesseq(X2,R2) ) ) ) ) ) ).

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