# 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, 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 = 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())

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..

In [5]:
stored_expr.style_options()

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

core typesub-expressionsexpression
0Lambdaparameter: 25
body: 2
1ExprTuple25
2Conditionalvalue: 3
condition: 4
3Operationoperator: 5
operand: 8
4Operationoperator: 26
operands: 7
5Literal
6ExprTuple8
7ExprTuple25, 9
8Lambdaparameters: 19
body: 10
9Literal
10Conditionalvalue: 11
condition: 12
11Operationoperator: 26
operands: 13
12Operationoperator: 14
operands: 15
13ExprTuple16, 29
14Literal
15ExprTuple17
16Operationoperator: 18
operands: 19
17ExprRangelambda_map: 20
start_index: 24
end_index: 25
18Literal
19ExprTuple21
20Lambdaparameter: 32
body: 22
21ExprRangelambda_map: 23
start_index: 24
end_index: 25
22Operationoperator: 26
operands: 27
23Lambdaparameter: 32
body: 28
24Literal
25Variable
26Literal
27ExprTuple28, 29
28IndexedVarvariable: 30
index: 32
29Literal
30Variable
31ExprTuple32
32Variable