# from the theory of proveit.numbers.exponentiation¶

In [1]:
import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
# import Expression classes needed to build the expression
from proveit import a, m, n
from proveit.logic import Equals, Forall
from proveit.numbers import Add, Complex, Exp, Mult, NaturalPos

In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [a, m, n], instance_expr = Equals(Mult(Exp(a, m), Exp(a, n)), Exp(a, Add(m, n))), domains = [Complex, NaturalPos, NaturalPos])

expr:
In [3]:
# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

In [4]:
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{a \in \mathbb{C}, m \in \mathbb{N}^+, n \in \mathbb{N}^+}~\left(\left(a^{m} \cdot a^{n}\right) = a^{m + n}\right)

In [5]:
stored_expr.style_options()

namedescriptiondefaultcurrent valuerelated methods
with_wrappingIf 'True', wrap the Expression after the parametersNoneNone/False('with_wrapping',)
condition_wrappingWrap 'before' or 'after' the condition (or None).NoneNone/False('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_paramsIf 'True', wraps every two parameters AND wraps the Expression after the parametersNoneNone/False('with_params',)
justificationjustify to the 'left', 'center', or 'right' in the array cellscentercenter('with_justification',)
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Operationoperator: 1
operand: 3
1Literal
2ExprTuple3
3Lambdaparameters: 4
body: 5
4ExprTuple34, 35, 36
5Conditionalvalue: 6
condition: 7
6Operationoperator: 8
operands: 9
7Operationoperator: 10
operands: 11
8Literal
9ExprTuple12, 13
10Literal
11ExprTuple14, 15, 16
12Operationoperator: 17
operands: 18
13Operationoperator: 30
operands: 19
14Operationoperator: 22
operands: 20
15Operationoperator: 22
operands: 21
16Operationoperator: 22
operands: 23
17Literal
18ExprTuple24, 25
19ExprTuple34, 26
20ExprTuple34, 27
21ExprTuple35, 28
22Literal
23ExprTuple36, 28
24Operationoperator: 30
operands: 29
25Operationoperator: 30
operands: 31
26Operationoperator: 32
operands: 33
27Literal
28Literal
29ExprTuple34, 35
30Literal
31ExprTuple34, 36
32Literal
33ExprTuple35, 36
34Variable
35Variable
36Variable