# 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.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import 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 = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [a_1_to_n], instance_expr = InSet(Mult(a_1_to_n), RationalNonNeg), domain = RationalNonNeg), 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} \in \mathbb{Q}^{\geq 0}}~\left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{n}\right) \in \mathbb{Q}^{\geq 0}\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: 27
body: 4
3ExprTuple27
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 28
operands: 9
7Literal
8ExprTuple10
9ExprTuple27, 11
10Lambdaparameters: 21
body: 12
11Literal
12Conditionalvalue: 13
condition: 14
13Operationoperator: 28
operands: 15
14Operationoperator: 16
operands: 17
15ExprTuple18, 31
16Literal
17ExprTuple19
18Operationoperator: 20
operands: 21
19ExprRangelambda_map: 22
start_index: 26
end_index: 27
20Literal
21ExprTuple23
22Lambdaparameter: 34
body: 24
23ExprRangelambda_map: 25
start_index: 26
end_index: 27
24Operationoperator: 28
operands: 29
25Lambdaparameter: 34
body: 30
26Literal
27Variable
28Literal
29ExprTuple30, 31
30IndexedVarvariable: 32
index: 34
31Literal
32Variable
33ExprTuple34
34Variable