# 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 Function, Q, f
from proveit.core_expr_types import a_1_to_i, b_1_to_j, c_1_to_k
from proveit.logic import Equals
from proveit.numbers import Mult, Sum
from proveit.numbers.summation import summation_b1toj_fQ

In [2]:
# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
expr = Equals(Mult(a_1_to_i, summation_b1toj_fQ, c_1_to_k), Sum(index_or_indices = sub_expr1, summand = Mult(a_1_to_i, Function(f, sub_expr1), c_1_to_k), condition = Function(Q, sub_expr1))).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right]\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) =  \\ \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right)\right] \end{array} \end{array}

In [5]:
stored_expr.style_options()

namedescriptiondefaultcurrent valuerelated methods
operation'infix' or 'function' style formattinginfixinfix
wrap_positionsposition(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.()(2)('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justificationif any wrap positions are set, justify to the 'left', 'center', or 'right'centercenter('with_justification',)
directionDirection of the relation (normal or reversed)normalnormal('with_direction_reversed', 'is_reversed')
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Operationoperator: 1
operands: 2
1Literal
2ExprTuple3, 4
3Operationoperator: 15
operands: 5
4Operationoperator: 9
operand: 8
5ExprTuple18, 7, 20
6ExprTuple8
7Operationoperator: 9
operand: 12
8Lambdaparameters: 25
body: 11
9Literal
10ExprTuple12
11Conditionalvalue: 13
condition: 17
12Lambdaparameters: 25
body: 14
13Operationoperator: 15
operands: 16
14Conditionalvalue: 19
condition: 17
15Literal
16ExprTuple18, 19, 20
17Operationoperator: 21
operands: 25
18ExprRangelambda_map: 22
start_index: 33
end_index: 23
19Operationoperator: 24
operands: 25
20ExprRangelambda_map: 26
start_index: 33
end_index: 27
21Variable
22Lambdaparameter: 39
body: 28
23Variable
24Variable
25ExprTuple29
26Lambdaparameter: 39
body: 30
27Variable
28IndexedVarvariable: 31
index: 39
29ExprRangelambda_map: 32
start_index: 33
end_index: 34
30IndexedVarvariable: 35
index: 39
31Variable
32Lambdaparameter: 39
body: 36
33Literal
34Variable
35Variable
36IndexedVarvariable: 37
index: 39
37Variable
38ExprTuple39
39Variable