# 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 Conditional, ExprTuple, Lambda, n
from proveit.core_expr_types import a_1_to_n
from proveit.logic import Forall, InSet
from proveit.numbers import Mult, Natural, RationalNonNeg

In [2]:
# build up the expression from sub-expressions
expr = ExprTuple(Lambda(n, Conditional(Forall(instance_param_or_params = [a_1_to_n], instance_expr = InSet(Mult(a_1_to_n), RationalNonNeg), domain = RationalNonNeg), InSet(n, 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())

\left(n \mapsto \left\{\forall_{a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{Q}^{\geq 0}}~\left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{n}\right) \in \mathbb{Q}^{\geq 0}\right) \textrm{ if } n \in \mathbb{N}\right..\right)

In [5]:
stored_expr.style_options()

no style options
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0ExprTuple1
1Lambdaparameter: 26
body: 3
2ExprTuple26
3Conditionalvalue: 4
condition: 5
4Operationoperator: 6
operand: 9
5Operationoperator: 27
operands: 8
6Literal
7ExprTuple9
8ExprTuple26, 10
9Lambdaparameters: 20
body: 11
10Literal
11Conditionalvalue: 12
condition: 13
12Operationoperator: 27
operands: 14
13Operationoperator: 15
operands: 16
14ExprTuple17, 30
15Literal
16ExprTuple18
17Operationoperator: 19
operands: 20
18ExprRangelambda_map: 21
start_index: 25
end_index: 26
19Literal
20ExprTuple22
21Lambdaparameter: 33
body: 23
22ExprRangelambda_map: 24
start_index: 25
end_index: 26
23Operationoperator: 27
operands: 28
24Lambdaparameter: 33
body: 29
25Literal
26Variable
27Literal
28ExprTuple29, 30
29IndexedVarvariable: 31
index: 33
30Literal
31Variable
32ExprTuple33
33Variable