## TPTP Problem File: SWW604=2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SWW604=2 : TPTP v8.0.0. Released v6.1.0.
% Domain   : Software Verification
% Problem  : Hashtbl impl-T-WP parameter clear
% 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    : hashtbl_impl-T-WP_parameter_clear [Fil14]

% Status   : Theorem
% Rating   : 0.50 v7.5.0, 0.60 v7.4.0, 0.50 v7.3.0, 0.33 v7.0.0, 0.71 v6.4.0, 0.33 v6.3.0, 0.71 v6.2.0, 0.62 v6.1.0
% Syntax   : Number of formulae    :  187 (  63 unt;  80 typ;   0 def)
%            Number of atoms       :  208 (  77 equ)
%            Maximal formula atoms :   27 (   1 avg)
%            Number of connectives :  112 (  11   ~;   5   |;  35   &)
%                                         (   5 <=>;  56  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   4 avg)
%            Maximal term depth    :    6 (   1 avg)
%            Number arithmetic     :  187 (  58 atm;  16 fun;  54 num;  59 var)
%            Number of types       :   13 (  11 usr;   1 ari)
%            Number of type conns  :  126 (  59   >;  67   *;   0   +;   0  <<)
%            Number of predicates  :    8 (   5 usr;   0 prp; 2-5 aty)
%            Number of functors    :   69 (  64 usr;  12 con; 0-5 aty)
%            Number of variables   :  274 ( 274   !;   0   ?; 274   :)
% 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(abs,type,
abs1: \$int > \$int ).

tff(abs_def,axiom,
! [X: \$int] :
( ( \$lesseq(0,X)
=> ( abs1(X) = X ) )
& ( ~ \$lesseq(0,X)
=> ( abs1(X) = \$uminus(X) ) ) ) ).

tff(abs_le,axiom,
! [X: \$int,Y: \$int] :
( \$lesseq(abs1(X),Y)
<=> ( \$lesseq(\$uminus(Y),X)
& \$lesseq(X,Y) ) ) ).

tff(abs_pos,axiom,
! [X: \$int] : \$lesseq(0,abs1(X)) ).

tff(div,type,
div1: ( \$int * \$int ) > \$int ).

tff(mod,type,
mod1: ( \$int * \$int ) > \$int ).

tff(div_mod,axiom,
! [X: \$int,Y: \$int] :
( ( Y != 0 )
=> ( X = \$sum(\$product(Y,div1(X,Y)),mod1(X,Y)) ) ) ).

tff(div_bound,axiom,
! [X: \$int,Y: \$int] :
( ( \$lesseq(0,X)
& \$less(0,Y) )
=> ( \$lesseq(0,div1(X,Y))
& \$lesseq(div1(X,Y),X) ) ) ).

tff(mod_bound,axiom,
! [X: \$int,Y: \$int] :
( ( Y != 0 )
=> ( \$less(\$uminus(abs1(Y)),mod1(X,Y))
& \$less(mod1(X,Y),abs1(Y)) ) ) ).

tff(div_sign_pos,axiom,
! [X: \$int,Y: \$int] :
( ( \$lesseq(0,X)
& \$less(0,Y) )
=> \$lesseq(0,div1(X,Y)) ) ).

tff(div_sign_neg,axiom,
! [X: \$int,Y: \$int] :
( ( \$lesseq(X,0)
& \$less(0,Y) )
=> \$lesseq(div1(X,Y),0) ) ).

tff(mod_sign_pos,axiom,
! [X: \$int,Y: \$int] :
( ( \$lesseq(0,X)
& ( Y != 0 ) )
=> \$lesseq(0,mod1(X,Y)) ) ).

tff(mod_sign_neg,axiom,
! [X: \$int,Y: \$int] :
( ( \$lesseq(X,0)
& ( Y != 0 ) )
=> \$lesseq(mod1(X,Y),0) ) ).

tff(rounds_toward_zero,axiom,
! [X: \$int,Y: \$int] :
( ( Y != 0 )
=> \$lesseq(abs1(\$product(div1(X,Y),Y)),abs1(X)) ) ).

tff(div_1,axiom,
! [X: \$int] : ( div1(X,1) = X ) ).

tff(mod_1,axiom,
! [X: \$int] : ( mod1(X,1) = 0 ) ).

tff(div_inf,axiom,
! [X: \$int,Y: \$int] :
( ( \$lesseq(0,X)
& \$less(X,Y) )
=> ( div1(X,Y) = 0 ) ) ).

tff(mod_inf,axiom,
! [X: \$int,Y: \$int] :
( ( \$lesseq(0,X)
& \$less(X,Y) )
=> ( mod1(X,Y) = X ) ) ).

tff(div_mult,axiom,
! [X: \$int,Y: \$int,Z: \$int] :
( ( \$less(0,X)
& \$lesseq(0,Y)
& \$lesseq(0,Z) )
=> ( div1(\$sum(\$product(X,Y),Z),X) = \$sum(Y,div1(Z,X)) ) ) ).

tff(mod_mult,axiom,
! [X: \$int,Y: \$int,Z: \$int] :
( ( \$less(0,X)
& \$lesseq(0,Y)
& \$lesseq(0,Z) )
=> ( mod1(\$sum(\$product(X,Y),Z),X) = mod1(Z,X) ) ) ).

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

tff(none,type,
none: ty > uni ).

tff(none_sort1,axiom,
! [A: ty] : sort1(option(A),none(A)) ).

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

tff(some_sort1,axiom,
! [A: ty,X: uni] : sort1(option(A),some(A,X)) ).

tff(match_option,type,
match_option1: ( ty * ty * uni * uni * uni ) > uni ).

tff(match_option_sort1,axiom,
! [A: ty,A1: ty,X: uni,X1: uni,X2: uni] : sort1(A1,match_option1(A1,A,X,X1,X2)) ).

tff(match_option_None1,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni] :
( sort1(A1,Z)
=> ( match_option1(A1,A,none(A),Z,Z1) = Z ) ) ).

tff(match_option_Some1,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni,U: uni] :
( sort1(A1,Z1)
=> ( match_option1(A1,A,some(A,U),Z,Z1) = Z1 ) ) ).

tff(none_Some1,axiom,
! [A: ty,V: uni] : ( none(A) != some(A,V) ) ).

tff(some_proj_1,type,
some_proj_11: ( ty * uni ) > uni ).

tff(some_proj_1_sort1,axiom,
! [A: ty,X: uni] : sort1(A,some_proj_11(A,X)) ).

tff(some_proj_1_def1,axiom,
! [A: ty,U: uni] :
( sort1(A,U)
=> ( some_proj_11(A,some(A,U)) = U ) ) ).

tff(option_inversion1,axiom,
! [A: ty,U: uni] :
( sort1(option(A),U)
=> ( ( U = none(A) )
| ( U = some(A,some_proj_11(A,U)) ) ) ) ).

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

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

tff(nil_sort1,axiom,
! [A: ty] : sort1(list(A),nil(A)) ).

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

tff(cons_sort1,axiom,
! [A: ty,X: uni,X1: uni] : sort1(list(A),cons(A,X,X1)) ).

tff(match_list,type,
match_list1: ( ty * ty * uni * uni * uni ) > uni ).

tff(match_list_sort1,axiom,
! [A: ty,A1: ty,X: uni,X1: uni,X2: uni] : sort1(A1,match_list1(A1,A,X,X1,X2)) ).

tff(match_list_Nil1,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni] :
( sort1(A1,Z)
=> ( match_list1(A1,A,nil(A),Z,Z1) = Z ) ) ).

tff(match_list_Cons1,axiom,
! [A: ty,A1: ty,Z: uni,Z1: uni,U: uni,U1: uni] :
( sort1(A1,Z1)
=> ( match_list1(A1,A,cons(A,U,U1),Z,Z1) = Z1 ) ) ).

tff(nil_Cons1,axiom,
! [A: ty,V: uni,V1: uni] : ( nil(A) != cons(A,V,V1) ) ).

tff(cons_proj_1,type,
cons_proj_11: ( ty * uni ) > uni ).

tff(cons_proj_1_sort1,axiom,
! [A: ty,X: uni] : sort1(A,cons_proj_11(A,X)) ).

tff(cons_proj_1_def1,axiom,
! [A: ty,U: uni,U1: uni] :
( sort1(A,U)
=> ( cons_proj_11(A,cons(A,U,U1)) = U ) ) ).

tff(cons_proj_2,type,
cons_proj_21: ( ty * uni ) > uni ).

tff(cons_proj_2_sort1,axiom,
! [A: ty,X: uni] : sort1(list(A),cons_proj_21(A,X)) ).

tff(cons_proj_2_def1,axiom,
! [A: ty,U: uni,U1: uni] : ( cons_proj_21(A,cons(A,U,U1)) = U1 ) ).

tff(list_inversion1,axiom,
! [A: ty,U: uni] :
( ( U = nil(A) )
| ( U = cons(A,cons_proj_11(A,U),cons_proj_21(A,U)) ) ) ).

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

tff(mem_def,axiom,
! [A: ty,X: uni] :
( sort1(A,X)
=> ( ~ mem(A,X,nil(A))
& ! [X1: uni,X2: uni] :
( sort1(A,X1)
=> ( mem(A,X,cons(A,X1,X2))
<=> ( ( X = X1 )
| mem(A,X,X2) ) ) ) ) ) ).

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

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

tff(get_sort2,axiom,
! [A: ty,B: ty,X: uni,X1: uni] : sort1(B,get(B,A,X,X1)) ).

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

tff(set_sort2,axiom,
! [A: ty,B: ty,X: uni,X1: uni,X2: uni] : sort1(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] :
( sort1(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] :
( sort1(A,A1)
=> ( sort1(A,A2)
=> ! [B1: uni] :
( ( A1 != A2 )
=> ( get(B,A,set(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(array,type,
array: ty > ty ).

tff(mk_array,type,
mk_array1: ( ty * \$int * uni ) > uni ).

tff(mk_array_sort1,axiom,
! [A: ty,X: \$int,X1: uni] : sort1(array(A),mk_array1(A,X,X1)) ).

tff(length,type,
length1: ( ty * uni ) > \$int ).

tff(length_def1,axiom,
! [A: ty,U: \$int,U1: uni] : ( length1(A,mk_array1(A,U,U1)) = U ) ).

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

tff(elts_sort1,axiom,
! [A: ty,X: uni] : sort1(map(int,A),elts(A,X)) ).

tff(elts_def1,axiom,
! [A: ty,U: \$int,U1: uni] :
( sort1(map(int,A),U1)
=> ( elts(A,mk_array1(A,U,U1)) = U1 ) ) ).

tff(array_inversion1,axiom,
! [A: ty,U: uni] : ( U = mk_array1(A,length1(A,U),elts(A,U)) ) ).

tff(get1,type,
get2: ( ty * uni * \$int ) > uni ).

tff(get_sort3,axiom,
! [A: ty,X: uni,X1: \$int] : sort1(A,get2(A,X,X1)) ).

tff(t2tb,type,
t2tb: \$int > uni ).

tff(t2tb_sort,axiom,
! [X: \$int] : sort1(int,t2tb(X)) ).

tff(tb2t,type,
tb2t: uni > \$int ).

tff(bridgeL,axiom,
! [I: \$int] : ( tb2t(t2tb(I)) = I ) ).

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

tff(get_def,axiom,
! [A: ty,A1: uni,I: \$int] : ( get2(A,A1,I) = get(A,int,elts(A,A1),t2tb(I)) ) ).

tff(set1,type,
set2: ( ty * uni * \$int * uni ) > uni ).

tff(set_sort3,axiom,
! [A: ty,X: uni,X1: \$int,X2: uni] : sort1(array(A),set2(A,X,X1,X2)) ).

tff(set_def,axiom,
! [A: ty,A1: uni,I: \$int,V: uni] : ( set2(A,A1,I,V) = mk_array1(A,length1(A,A1),set(A,int,elts(A,A1),t2tb(I),V)) ) ).

tff(make,type,
make1: ( ty * \$int * uni ) > uni ).

tff(make_sort1,axiom,
! [A: ty,X: \$int,X1: uni] : sort1(array(A),make1(A,X,X1)) ).

tff(make_def,axiom,
! [A: ty,N: \$int,V: uni] : ( make1(A,N,V) = mk_array1(A,N,const(A,int,V)) ) ).

tff(key,type,
key1: \$tType ).

tff(key1,type,
key: ty ).

tff(hash,type,
hash1: key1 > \$int ).

tff(hash_nonneg,axiom,
! [K: key1] : \$lesseq(0,hash1(K)) ).

tff(bucket,type,
bucket1: ( key1 * \$int ) > \$int ).

tff(bucket_def,axiom,
! [K: key1,N: \$int] : ( bucket1(K,N) = mod1(hash1(K),N) ) ).

tff(bucket_bounds,axiom,
! [N: \$int] :
( \$less(0,N)
=> ! [K: key1] :
( \$lesseq(0,bucket1(K,N))
& \$less(bucket1(K,N),N) ) ) ).

tff(tuple2,type,
tuple2: ( ty * ty ) > ty ).

tff(tuple21,type,
tuple21: ( ty * ty * uni * uni ) > uni ).

tff(tuple2_sort1,axiom,
! [A: ty,A1: ty,X: uni,X1: uni] : sort1(tuple2(A1,A),tuple21(A1,A,X,X1)) ).

tff(tuple2_proj_1,type,
tuple2_proj_11: ( ty * ty * uni ) > uni ).

tff(tuple2_proj_1_sort1,axiom,
! [A: ty,A1: ty,X: uni] : sort1(A1,tuple2_proj_11(A1,A,X)) ).

tff(tuple2_proj_1_def1,axiom,
! [A: ty,A1: ty,U: uni,U1: uni] :
( sort1(A1,U)
=> ( tuple2_proj_11(A1,A,tuple21(A1,A,U,U1)) = U ) ) ).

tff(tuple2_proj_2,type,
tuple2_proj_21: ( ty * ty * uni ) > uni ).

tff(tuple2_proj_2_sort1,axiom,
! [A: ty,A1: ty,X: uni] : sort1(A,tuple2_proj_21(A1,A,X)) ).

tff(tuple2_proj_2_def1,axiom,
! [A: ty,A1: ty,U: uni,U1: uni] :
( sort1(A,U1)
=> ( tuple2_proj_21(A1,A,tuple21(A1,A,U,U1)) = U1 ) ) ).

tff(tuple2_inversion1,axiom,
! [A: ty,A1: ty,U: uni] :
( sort1(tuple2(A1,A),U)
=> ( U = tuple21(A1,A,tuple2_proj_11(A1,A,U),tuple2_proj_21(A1,A,U)) ) ) ).

tff(in_data,type,
in_data1: ( ty * key1 * uni * uni ) > \$o ).

tff(t2tb1,type,
t2tb1: key1 > uni ).

tff(t2tb_sort1,axiom,
! [X: key1] : sort1(key,t2tb1(X)) ).

tff(tb2t1,type,
tb2t1: uni > key1 ).

tff(bridgeL1,axiom,
! [I: key1] : ( tb2t1(t2tb1(I)) = I ) ).

tff(bridgeR1,axiom,
! [J: uni] :
( sort1(key,J)
=> ( t2tb1(tb2t1(J)) = J ) ) ).

tff(in_data_def,axiom,
! [A: ty,K: key1,V: uni,D: uni] :
( in_data1(A,K,V,D)
<=> mem(tuple2(key,A),tuple21(key,A,t2tb1(K),V),get2(list(tuple2(key,A)),D,bucket1(K,length1(list(tuple2(key,A)),D)))) ) ).

tff(good_data,type,
good_data1: ( ty * key1 * uni * uni * uni ) > \$o ).

tff(good_data_def,axiom,
! [A: ty,K: key1,V: uni,M: uni,D: uni] :
( good_data1(A,K,V,M,D)
<=> ( ( get(option(A),key,M,t2tb1(K)) = some(A,V) )
<=> in_data1(A,K,V,D) ) ) ).

tff(good_hash,type,
good_hash1: ( ty * uni * \$int ) > \$o ).

tff(good_hash_def,axiom,
! [A: ty,D: uni,I: \$int] :
( ( good_hash1(A,D,I)
=> ! [K: key1,V: uni] :
( mem(tuple2(key,A),tuple21(key,A,t2tb1(K),V),get2(list(tuple2(key,A)),D,I))
=> ( bucket1(K,length1(list(tuple2(key,A)),D)) = I ) ) )
& ( ! [K: key1,V: uni] :
( sort1(A,V)
=> ( mem(tuple2(key,A),tuple21(key,A,t2tb1(K),V),get2(list(tuple2(key,A)),D,I))
=> ( bucket1(K,length1(list(tuple2(key,A)),D)) = I ) ) )
=> good_hash1(A,D,I) ) ) ).

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

tff(mk_t,type,
mk_t1: ( ty * \$int * uni * uni ) > uni ).

tff(mk_t_sort1,axiom,
! [A: ty,X: \$int,X1: uni,X2: uni] : sort1(t(A),mk_t1(A,X,X1,X2)) ).

tff(size,type,
size1: ( ty * uni ) > \$int ).

tff(size_def1,axiom,
! [A: ty,U: \$int,U1: uni,U2: uni] : ( size1(A,mk_t1(A,U,U1,U2)) = U ) ).

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

tff(data_sort1,axiom,
! [A: ty,X: uni] : sort1(array(list(tuple2(key,A))),data(A,X)) ).

tff(data_def1,axiom,
! [A: ty,U: \$int,U1: uni,U2: uni] : ( data(A,mk_t1(A,U,U1,U2)) = U1 ) ).

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

tff(view_sort1,axiom,
! [A: ty,X: uni] : sort1(map(key,option(A)),view(A,X)) ).

tff(view_def1,axiom,
! [A: ty,U: \$int,U1: uni,U2: uni] :
( sort1(map(key,option(A)),U2)
=> ( view(A,mk_t1(A,U,U1,U2)) = U2 ) ) ).

tff(t_inversion1,axiom,
! [A: ty,U: uni] : ( U = mk_t1(A,size1(A,U),data(A,U),view(A,U)) ) ).

tff(a,type,
a1: \$tType ).

tff(a1,type,
a: ty ).

tff(map_int_lplist_lpkeycm_a1rprp,type,
map_int_lplist_lpkeycm_a1rprp: \$tType ).

tff(t2tb2,type,
t2tb2: map_int_lplist_lpkeycm_a1rprp > uni ).

tff(t2tb_sort2,axiom,
! [X: map_int_lplist_lpkeycm_a1rprp] : sort1(map(int,list(tuple2(key,a))),t2tb2(X)) ).

tff(tb2t2,type,
tb2t2: uni > map_int_lplist_lpkeycm_a1rprp ).

tff(bridgeL2,axiom,
! [I: map_int_lplist_lpkeycm_a1rprp] : ( tb2t2(t2tb2(I)) = I ) ).

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

tff(map_key_lpoption_a1rp,type,
map_key_lpoption_a1rp: \$tType ).

tff(t2tb3,type,
t2tb3: map_key_lpoption_a1rp > uni ).

tff(t2tb_sort3,axiom,
! [X: map_key_lpoption_a1rp] : sort1(map(key,option(a)),t2tb3(X)) ).

tff(tb2t3,type,
tb2t3: uni > map_key_lpoption_a1rp ).

tff(bridgeL3,axiom,
! [I: map_key_lpoption_a1rp] : ( tb2t3(t2tb3(I)) = I ) ).

tff(bridgeR3,axiom,
! [J: uni] :
( sort1(map(key,option(a)),J)
=> ( t2tb3(tb2t3(J)) = J ) ) ).

tff(option_a1,type,
option_a1: \$tType ).

tff(t2tb4,type,
t2tb4: option_a1 > uni ).

tff(t2tb_sort4,axiom,
! [X: option_a1] : sort1(option(a),t2tb4(X)) ).

tff(tb2t4,type,
tb2t4: uni > option_a1 ).

tff(bridgeL4,axiom,
! [I: option_a1] : ( tb2t4(t2tb4(I)) = I ) ).

tff(bridgeR4,axiom,
! [J: uni] :
( sort1(option(a),J)
=> ( t2tb4(tb2t4(J)) = J ) ) ).

tff(array_lplist_lpkeycm_a1rprp,type,
array_lplist_lpkeycm_a1rprp: \$tType ).

tff(t2tb5,type,
t2tb5: array_lplist_lpkeycm_a1rprp > uni ).

tff(t2tb_sort5,axiom,
! [X: array_lplist_lpkeycm_a1rprp] : sort1(array(list(tuple2(key,a))),t2tb5(X)) ).

tff(tb2t5,type,
tb2t5: uni > array_lplist_lpkeycm_a1rprp ).

tff(bridgeL5,axiom,
! [I: array_lplist_lpkeycm_a1rprp] : ( tb2t5(t2tb5(I)) = I ) ).

tff(bridgeR5,axiom,
! [J: uni] : ( t2tb5(tb2t5(J)) = J ) ).

tff(list_lpkeycm_a1rp,type,
list_lpkeycm_a1rp: \$tType ).

tff(t2tb6,type,
t2tb6: list_lpkeycm_a1rp > uni ).

tff(t2tb_sort6,axiom,
! [X: list_lpkeycm_a1rp] : sort1(list(tuple2(key,a)),t2tb6(X)) ).

tff(tb2t6,type,
tb2t6: uni > list_lpkeycm_a1rp ).

tff(bridgeL6,axiom,
! [I: list_lpkeycm_a1rp] : ( tb2t6(t2tb6(I)) = I ) ).

tff(bridgeR6,axiom,
! [J: uni] : ( t2tb6(tb2t6(J)) = J ) ).

tff(t2tb7,type,
t2tb7: a1 > uni ).

tff(t2tb_sort7,axiom,
! [X: a1] : sort1(a,t2tb7(X)) ).

tff(tb2t7,type,
tb2t7: uni > a1 ).

tff(bridgeL7,axiom,
! [I: a1] : ( tb2t7(t2tb7(I)) = I ) ).

tff(bridgeR7,axiom,
! [J: uni] :
( sort1(a,J)
=> ( t2tb7(tb2t7(J)) = J ) ) ).

tff(wP_parameter_clear,conjecture,
! [H: \$int,H1: map_int_lplist_lpkeycm_a1rprp,H2: map_key_lpoption_a1rp] :
( ( \$less(0,H)
& ! [I: \$int] :
( ( \$lesseq(0,I)
& \$less(I,H) )
=> good_hash1(a,mk_array1(list(tuple2(key,a)),H,t2tb2(H1)),I) )
& ! [K: key1,V: a1] : good_data1(a,K,t2tb7(V),t2tb3(H2),mk_array1(list(tuple2(key,a)),H,t2tb2(H1)))
& \$lesseq(0,H) )
=> ! [Rho: \$int] :
( ( Rho = 0 )
=> ( \$lesseq(0,0)
& \$lesseq(0,H)
& \$lesseq(\$sum(0,H),H)
& ! [O: map_int_lplist_lpkeycm_a1rprp] :
( ( \$lesseq(0,H)
& ! [I: \$int] :
( ( ( \$lesseq(0,I)
& \$less(I,0) )
| ( \$lesseq(\$sum(0,H),I)
& \$less(I,H) ) )
=> ( tb2t6(get(list(tuple2(key,a)),int,t2tb2(O),t2tb(I))) = tb2t6(get(list(tuple2(key,a)),int,t2tb2(H1),t2tb(I))) ) )
& ! [I: \$int] :
( ( \$lesseq(0,I)
& \$less(I,\$sum(0,H)) )
=> ( tb2t6(get(list(tuple2(key,a)),int,t2tb2(O),t2tb(I))) = tb2t6(nil(tuple2(key,a))) ) ) )
=> ! [Rho1: map_key_lpoption_a1rp] :
( ( Rho1 = tb2t3(const(option(a),key,none(a))) )
=> ( \$less(0,H)
& ! [I: \$int] :
( ( \$lesseq(0,I)
& \$less(I,H) )
=> good_hash1(a,mk_array1(list(tuple2(key,a)),H,t2tb2(O)),I) )
& ! [K: key1,V: a1] : good_data1(a,K,t2tb7(V),t2tb3(Rho1),mk_array1(list(tuple2(key,a)),H,t2tb2(O)))
& \$lesseq(0,H)
& ( Rho1 = tb2t3(const(option(a),key,none(a))) ) ) ) ) ) ) ) ).

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