# from the theory of proveit.numbers.multiplication¶

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 b, n
from proveit.core_expr_types import a_1_to_n
from proveit.logic import Equals, Forall
from proveit.numbers import Complex, Mult, Natural

In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [a_1_to_n, b], instance_expr = Equals(Mult(a_1_to_n, b), Mult(Mult(a_1_to_n), b)), domain = Complex), domain = Natural)

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_{n \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{n}, b \in \mathbb{C}}~\left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{n} \cdot b\right) = \left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{n}\right) \cdot b\right)\right)\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: 7
operand: 2
1ExprTuple2
2Lambdaparameter: 38
body: 4
3ExprTuple38
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 32
operands: 9
7Literal
8ExprTuple10
9ExprTuple38, 11
10Lambdaparameters: 23
body: 12
11Literal
12Conditionalvalue: 13
condition: 14
13Operationoperator: 15
operands: 16
14Operationoperator: 17
operands: 18
15Literal
16ExprTuple19, 20
17Literal
18ExprTuple21, 22
19Operationoperator: 30
operands: 23
20Operationoperator: 30
operands: 24
21ExprRangelambda_map: 25
start_index: 37
end_index: 38
22Operationoperator: 32
operands: 26
23ExprTuple34, 29
24ExprTuple27, 29
25Lambdaparameter: 42
body: 28
26ExprTuple29, 35
27Operationoperator: 30
operands: 31
28Operationoperator: 32
operands: 33
29Variable
30Literal
31ExprTuple34
32Literal
33ExprTuple39, 35
34ExprRangelambda_map: 36
start_index: 37
end_index: 38
35Literal
36Lambdaparameter: 42
body: 39
37Literal
38Variable
39IndexedVarvariable: 40
index: 42
40Variable
41ExprTuple42
42Variable