In [1]:

```
import proveit
from proveit import defaults
from proveit import a, m, n, x, A, B
from proveit.core_expr_types import A_1_to_m
from proveit.numbers import zero, num, Neg, NaturalPos, Integer, greater, Natural, greater_eq
from proveit.logic import NotInSet, SetOfAll, InSet, Implies, And
from proveit.logic.sets.unification import membership_unfolding
from proveit.numbers.number_sets.natural_numbers import natural_pos_def
from proveit.numbers.number_sets.integers import integers_def
theory = proveit.Theory() # the theorem's theory
```

In [2]:

```
%proving nonneg_int_is_natural
```

In [3]:

```
# defaults.assumptions = nonneg_int_is_natural.conditions
```

In [4]:

```
# integers_def
```

In [5]:

```
# int_union = integers_def.expr.rhs
```

In [6]:

```
# int_union.operands
```

In [7]:

```
# membership_unfolding
```

In [8]:

```
# spec1 = membership_unfolding.instantiate({x:a, m:num(2), A:int_union.operands}, assumptions = [InSet(m, NaturalPos), *defaults.assumptions])
```

In [9]:

```
# a_in_int = defaults.assumptions[0]
```

In [10]:

```
# a_elem_union = Implies(And(a_in_int, integers_def.expr),InSet(a,int_union))
```

In [11]:

```
# new_assumpt = [defaults.assumptions]+ [integers_def.expr]
```

These web pages were generated on 2023-08-23 by Prove-It Beta Version 0.3, licensed under the GNU Public License by Sandia Corporation.

Presented proofs are not absolutely guaranteed. For assurance, it is important to check the structure of the statement being proven, independently verify the derivation steps, track dependencies, and ensure that employed axioms are valid and properly structured. Inconsistencies may exist, unknowingly, in this system.

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, under the Quantum Computing Application Teams program. Sandia National Labs is managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a subsidiary of Honeywell International, Inc., for the U.S. Dept. of Energy's NNSA under contract DE-NA0003525. The views expressed above do not necessarily represent the views of the DOE or the U.S. Government.

Please send questions/comments to: wwitzel@sandia.gov.