## TPTP Problem File: SWW620=2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SWW620=2 : TPTP v8.0.0. Released v6.1.0.
% Domain   : Software Verification
% Problem  : Mergesort array-T-WP parameter find run
% 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    : mergesort_array-T-WP_parameter_find_run [Fil14]

% Status   : Theorem
% Rating   : 0.38 v7.5.0, 0.40 v7.4.0, 0.12 v7.3.0, 0.00 v7.0.0, 0.14 v6.4.0, 0.00 v6.3.0, 0.43 v6.2.0, 0.50 v6.1.0
% Syntax   : Number of formulae    :  157 (  33 unt;  59 typ;   0 def)
%            Number of atoms       :  254 (  76 equ)
%            Maximal formula atoms :   12 (   1 avg)
%            Number of connectives :  168 (  12   ~;   4   |;  63   &)
%                                         (  11 <=>;  78  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   20 (   6 avg)
%            Maximal term depth    :    5 (   1 avg)
%            Number arithmetic     :  321 ( 107 atm;  21 fun;  58 num; 135 var)
%            Number of types       :    9 (   7 usr;   1 ari)
%            Number of type conns  :  115 (  43   >;  72   *;   0   +;   0  <<)
%            Number of predicates  :   16 (  13 usr;   0 prp; 1-7 aty)
%            Number of functors    :   45 (  39 usr;  11 con; 0-5 aty)
%            Number of variables   :  301 ( 299   !;   2   ?; 301   :)
% 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_sort4,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(map,type,
map: ( ty * ty ) > ty ).

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

tff(get_sort8,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_sort8,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_sort4,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_sort4,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_sort4,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_sort9,axiom,
! [A: ty,X: uni,X1: \$int] : sort1(A,get2(A,X,X1)) ).

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

tff(t2tb_sort12,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_sort9,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_sort4,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(elt6,type,
elt6: \$tType ).

tff(elt7,type,
elt7: ty ).

tff(le3,type,
le4: ( elt6 * elt6 ) > \$o ).

tff(refl4,axiom,
! [X: elt6] : le4(X,X) ).

tff(trans4,axiom,
! [X: elt6,Y: elt6,Z: elt6] :
( le4(X,Y)
=> ( le4(Y,Z)
=> le4(X,Z) ) ) ).

tff(total4,axiom,
! [X: elt6,Y: elt6] :
( le4(X,Y)
| le4(Y,X) ) ).

tff(array_elt3,type,
array_elt3: \$tType ).

tff(sorted_sub3,type,
sorted_sub4: ( array_elt3 * \$int * \$int ) > \$o ).

tff(t2tb10,type,
t2tb10: array_elt3 > uni ).

tff(t2tb_sort13,axiom,
! [X: array_elt3] : sort1(array(elt7),t2tb10(X)) ).

tff(tb2t10,type,
tb2t10: uni > array_elt3 ).

tff(bridgeL10,axiom,
! [I: array_elt3] : ( tb2t10(t2tb10(I)) = I ) ).

tff(bridgeR10,axiom,
! [J: uni] : ( t2tb10(tb2t10(J)) = J ) ).

tff(t2tb11,type,
t2tb11: elt6 > uni ).

tff(t2tb_sort14,axiom,
! [X: elt6] : sort1(elt7,t2tb11(X)) ).

tff(tb2t11,type,
tb2t11: uni > elt6 ).

tff(bridgeL11,axiom,
! [I: elt6] : ( tb2t11(t2tb11(I)) = I ) ).

tff(bridgeR11,axiom,
! [J: uni] :
( sort1(elt7,J)
=> ( t2tb11(tb2t11(J)) = J ) ) ).

tff(sorted_sub_def3,axiom,
! [A: array_elt3,L: \$int,U: \$int] :
( sorted_sub4(A,L,U)
<=> ! [I1: \$int,I2: \$int] :
( ( \$lesseq(L,I1)
& \$lesseq(I1,I2)
& \$less(I2,U) )
=> le4(tb2t11(get2(elt7,t2tb10(A),I1)),tb2t11(get2(elt7,t2tb10(A),I2))) ) ) ).

tff(sorted3,type,
sorted4: array_elt3 > \$o ).

tff(sorted_def3,axiom,
! [A: array_elt3] :
( sorted4(A)
<=> ! [I1: \$int,I2: \$int] :
( ( \$lesseq(0,I1)
& \$lesseq(I1,I2)
& \$less(I2,length1(elt7,t2tb10(A))) )
=> le4(tb2t11(get2(elt7,t2tb10(A),I1)),tb2t11(get2(elt7,t2tb10(A),I2))) ) ) ).

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

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

tff(mk_ref_sort4,axiom,
! [A: ty,X: uni] : sort1(ref(A),mk_ref(A,X)) ).

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

tff(contents_sort4,axiom,
! [A: ty,X: uni] : sort1(A,contents(A,X)) ).

tff(contents_def4,axiom,
! [A: ty,U: uni] :
( sort1(A,U)
=> ( contents(A,mk_ref(A,U)) = U ) ) ).

tff(ref_inversion4,axiom,
! [A: ty,U: uni] :
( sort1(ref(A),U)
=> ( U = mk_ref(A,contents(A,U)) ) ) ).

tff(occ,type,
occ1: ( ty * uni * uni * \$int * \$int ) > \$int ).

tff(occ_empty,axiom,
! [A: ty,V: uni,M: uni,L: \$int,U: \$int] :
( \$lesseq(U,L)
=> ( occ1(A,V,M,L,U) = 0 ) ) ).

! [A: ty,V: uni,M: uni,L: \$int,U: \$int] :
( sort1(A,V)
=> ( \$less(L,U)
=> ( ( get(A,int,M,t2tb(\$difference(U,1))) != V )
=> ( occ1(A,V,M,L,U) = occ1(A,V,M,L,\$difference(U,1)) ) ) ) ) ).

! [A: ty,V: uni,M: uni,L: \$int,U: \$int] :
( \$less(L,U)
=> ( ( get(A,int,M,t2tb(\$difference(U,1))) = V )
=> ( occ1(A,V,M,L,U) = \$sum(1,occ1(A,V,M,L,\$difference(U,1))) ) ) ) ).

tff(occ_bounds,axiom,
! [A: ty,V: uni,M: uni,L: \$int,U: \$int] :
( \$lesseq(L,U)
=> ( \$lesseq(0,occ1(A,V,M,L,U))
& \$lesseq(occ1(A,V,M,L,U),\$difference(U,L)) ) ) ).

tff(occ_append,axiom,
! [A: ty,V: uni,M: uni,L: \$int,Mid: \$int,U: \$int] :
( ( \$lesseq(L,Mid)
& \$lesseq(Mid,U) )
=> ( occ1(A,V,M,L,U) = \$sum(occ1(A,V,M,L,Mid),occ1(A,V,M,Mid,U)) ) ) ).

tff(occ_neq,axiom,
! [A: ty,V: uni,M: uni,L: \$int,U: \$int] :
( sort1(A,V)
=> ( ! [I: \$int] :
( ( \$lesseq(L,I)
& \$less(I,U) )
=> ( get(A,int,M,t2tb(I)) != V ) )
=> ( occ1(A,V,M,L,U) = 0 ) ) ) ).

tff(occ_exists,axiom,
! [A: ty,V: uni,M: uni,L: \$int,U: \$int] :
( sort1(A,V)
=> ( \$less(0,occ1(A,V,M,L,U))
=> ? [I: \$int] :
( \$lesseq(L,I)
& \$less(I,U)
& ( get(A,int,M,t2tb(I)) = V ) ) ) ) ).

tff(occ_pos,axiom,
! [A: ty,M: uni,L: \$int,U: \$int,I: \$int] :
( ( \$lesseq(L,I)
& \$less(I,U) )
=> \$less(0,occ1(A,get(A,int,M,t2tb(I)),M,L,U)) ) ).

tff(occ_eq,axiom,
! [A: ty,V: uni,M1: uni,M2: uni,L: \$int,U: \$int] :
( ! [I: \$int] :
( ( \$lesseq(L,I)
& \$less(I,U) )
=> ( get(A,int,M1,t2tb(I)) = get(A,int,M2,t2tb(I)) ) )
=> ( occ1(A,V,M1,L,U) = occ1(A,V,M2,L,U) ) ) ).

tff(permut,type,
permut2: ( ty * uni * uni * \$int * \$int ) > \$o ).

tff(permut_def,axiom,
! [A: ty,M1: uni,M2: uni,L: \$int,U: \$int] :
( ( permut2(A,M1,M2,L,U)
=> ! [V: uni] : ( occ1(A,V,M1,L,U) = occ1(A,V,M2,L,U) ) )
& ( ! [V: uni] :
( sort1(A,V)
=> ( occ1(A,V,M1,L,U) = occ1(A,V,M2,L,U) ) )
=> permut2(A,M1,M2,L,U) ) ) ).

tff(permut_trans,axiom,
! [A: ty,A1: uni,A2: uni,A3: uni,L: \$int,U: \$int] :
( permut2(A,A1,A2,L,U)
=> ( permut2(A,A2,A3,L,U)
=> permut2(A,A1,A3,L,U) ) ) ).

tff(permut_exists,axiom,
! [A: ty,A1: uni,A2: uni,L: \$int,U: \$int,I: \$int] :
( permut2(A,A1,A2,L,U)
=> ( ( \$lesseq(L,I)
& \$less(I,U) )
=> ? [J: \$int] :
( \$lesseq(L,J)
& \$less(J,U)
& ( get(A,int,A1,t2tb(J)) = get(A,int,A2,t2tb(I)) ) ) ) ) ).

tff(map_eq_sub,type,
map_eq_sub1: ( ty * uni * uni * \$int * \$int ) > \$o ).

tff(map_eq_sub_def,axiom,
! [A: ty,A1: uni,A2: uni,L: \$int,U: \$int] :
( map_eq_sub1(A,A1,A2,L,U)
<=> ! [I: \$int] :
( ( \$lesseq(L,I)
& \$less(I,U) )
=> ( get(A,int,A1,t2tb(I)) = get(A,int,A2,t2tb(I)) ) ) ) ).

tff(array_eq_sub,type,
array_eq_sub1: ( ty * uni * uni * \$int * \$int ) > \$o ).

tff(array_eq_sub_def,axiom,
! [A: ty,A1: uni,A2: uni,L: \$int,U: \$int] :
( array_eq_sub1(A,A1,A2,L,U)
<=> ( ( length1(A,A1) = length1(A,A2) )
& \$lesseq(0,L)
& \$lesseq(L,length1(A,A1))
& \$lesseq(0,U)
& \$lesseq(U,length1(A,A1))
& map_eq_sub1(A,elts(A,A1),elts(A,A2),L,U) ) ) ).

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

tff(array_eq_def,axiom,
! [A: ty,A1: uni,A2: uni] :
( array_eq(A,A1,A2)
<=> ( ( length1(A,A1) = length1(A,A2) )
& map_eq_sub1(A,elts(A,A1),elts(A,A2),0,length1(A,A1)) ) ) ).

tff(exchange,type,
exchange2: ( ty * uni * uni * \$int * \$int * \$int * \$int ) > \$o ).

tff(exchange_def,axiom,
! [A: ty,A1: uni,A2: uni,L: \$int,U: \$int,I: \$int,J: \$int] :
( exchange2(A,A1,A2,L,U,I,J)
<=> ( \$lesseq(L,I)
& \$less(I,U)
& \$lesseq(L,J)
& \$less(J,U)
& ( get(A,int,A1,t2tb(I)) = get(A,int,A2,t2tb(J)) )
& ( get(A,int,A1,t2tb(J)) = get(A,int,A2,t2tb(I)) )
& ! [K: \$int] :
( ( \$lesseq(L,K)
& \$less(K,U) )
=> ( ( K != I )
=> ( ( K != J )
=> ( get(A,int,A1,t2tb(K)) = get(A,int,A2,t2tb(K)) ) ) ) ) ) ) ).

tff(exchange_set,axiom,
! [A: ty,A1: uni,L: \$int,U: \$int,I: \$int,J: \$int] :
( ( \$lesseq(L,I)
& \$less(I,U) )
=> ( ( \$lesseq(L,J)
& \$less(J,U) )
=> exchange2(A,A1,set(A,int,set(A,int,A1,t2tb(I),get(A,int,A1,t2tb(J))),t2tb(J),get(A,int,A1,t2tb(I))),L,U,I,J) ) ) ).

tff(exchange1,type,
exchange3: ( ty * uni * uni * \$int * \$int ) > \$o ).

tff(exchange_def1,axiom,
! [A: ty,A1: uni,A2: uni,I: \$int,J: \$int] :
( exchange3(A,A1,A2,I,J)
<=> ( ( length1(A,A1) = length1(A,A2) )
& exchange2(A,elts(A,A1),elts(A,A2),0,length1(A,A1),I,J) ) ) ).

tff(permut1,type,
permut3: ( ty * uni * uni * \$int * \$int ) > \$o ).

tff(permut_def1,axiom,
! [A: ty,A1: uni,A2: uni,L: \$int,U: \$int] :
( permut3(A,A1,A2,L,U)
<=> ( ( length1(A,A1) = length1(A,A2) )
& \$lesseq(0,L)
& \$lesseq(L,length1(A,A1))
& \$lesseq(0,U)
& \$lesseq(U,length1(A,A1))
& permut2(A,elts(A,A1),elts(A,A2),L,U) ) ) ).

tff(permut_sub,type,
permut_sub1: ( ty * uni * uni * \$int * \$int ) > \$o ).

tff(permut_sub_def,axiom,
! [A: ty,A1: uni,A2: uni,L: \$int,U: \$int] :
( permut_sub1(A,A1,A2,L,U)
<=> ( map_eq_sub1(A,elts(A,A1),elts(A,A2),0,L)
& permut3(A,A1,A2,L,U)
& map_eq_sub1(A,elts(A,A1),elts(A,A2),U,length1(A,A1)) ) ) ).

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

tff(permut_all_def,axiom,
! [A: ty,A1: uni,A2: uni] :
( permut_all(A,A1,A2)
<=> ( ( length1(A,A1) = length1(A,A2) )
& permut2(A,elts(A,A1),elts(A,A2),0,length1(A,A1)) ) ) ).

tff(exchange_permut_sub,axiom,
! [A: ty,A1: uni,A2: uni,I: \$int,J: \$int,L: \$int,U: \$int] :
( exchange3(A,A1,A2,I,J)
=> ( ( \$lesseq(L,I)
& \$less(I,U) )
=> ( ( \$lesseq(L,J)
& \$less(J,U) )
=> ( \$lesseq(0,L)
=> ( \$lesseq(U,length1(A,A1))
=> permut_sub1(A,A1,A2,L,U) ) ) ) ) ) ).

tff(permut_sub_weakening,axiom,
! [A: ty,A1: uni,A2: uni,L1: \$int,U1: \$int,L2: \$int,U2: \$int] :
( permut_sub1(A,A1,A2,L1,U1)
=> ( ( \$lesseq(0,L2)
& \$lesseq(L2,L1) )
=> ( ( \$lesseq(U1,U2)
& \$lesseq(U2,length1(A,A1)) )
=> permut_sub1(A,A1,A2,L2,U2) ) ) ) ).

tff(exchange_permut_all,axiom,
! [A: ty,A1: uni,A2: uni,I: \$int,J: \$int] :
( exchange3(A,A1,A2,I,J)
=> permut_all(A,A1,A2) ) ).

tff(abs,type,
abs: \$int > \$int ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(max_is_ge,axiom,
! [X: \$int,Y: \$int] :
( \$lesseq(X,max(X,Y))
& \$lesseq(Y,max(X,Y)) ) ).

tff(max_is_some,axiom,
! [X: \$int,Y: \$int] :
( ( max(X,Y) = X )
| ( max(X,Y) = Y ) ) ).

tff(min_is_le,axiom,
! [X: \$int,Y: \$int] :
( \$lesseq(min(X,Y),X)
& \$lesseq(min(X,Y),Y) ) ).

tff(min_is_some,axiom,
! [X: \$int,Y: \$int] :
( ( min(X,Y) = X )
| ( min(X,Y) = Y ) ) ).

tff(max_x,axiom,
! [X: \$int,Y: \$int] :
( \$lesseq(Y,X)
=> ( max(X,Y) = X ) ) ).

tff(max_y,axiom,
! [X: \$int,Y: \$int] :
( \$lesseq(X,Y)
=> ( max(X,Y) = Y ) ) ).

tff(min_x,axiom,
! [X: \$int,Y: \$int] :
( \$lesseq(X,Y)
=> ( min(X,Y) = X ) ) ).

tff(min_y,axiom,
! [X: \$int,Y: \$int] :
( \$lesseq(Y,X)
=> ( min(X,Y) = Y ) ) ).

tff(max_sym,axiom,
! [X: \$int,Y: \$int] :
( \$lesseq(Y,X)
=> ( max(X,Y) = max(Y,X) ) ) ).

tff(min_sym,axiom,
! [X: \$int,Y: \$int] :
( \$lesseq(Y,X)
=> ( min(X,Y) = min(Y,X) ) ) ).

tff(map_int_elt3,type,
map_int_elt3: \$tType ).

tff(t2tb12,type,
t2tb12: map_int_elt3 > uni ).

tff(t2tb_sort15,axiom,
! [X: map_int_elt3] : sort1(map(int,elt7),t2tb12(X)) ).

tff(tb2t12,type,
tb2t12: uni > map_int_elt3 ).

tff(bridgeL12,axiom,
! [I: map_int_elt3] : ( tb2t12(t2tb12(I)) = I ) ).

tff(bridgeR12,axiom,
! [J: uni] :
( sort1(map(int,elt7),J)
=> ( t2tb12(tb2t12(J)) = J ) ) ).

tff(wP_parameter_find_run,conjecture,
! [A: \$int,A1: map_int_elt3,Lo: \$int] :
( ( \$lesseq(0,A)
& \$lesseq(0,Lo)
& \$less(Lo,A) )
=> sorted_sub4(tb2t10(mk_array1(elt7,A,t2tb12(A1))),Lo,\$sum(Lo,1)) ) ).

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