# from the theory of proveit.numbers.multiplication¶

see dependencies

In [1]:
import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
# import the special expression
from proveit.numbers.multiplication import rightward_commutation

In [2]:
# check that the built expression is the same as the stored expression
assert rightward_commutation.expr == stored_expr
assert rightward_commutation.expr._style_id == stored_expr._style_id
print("Passed sanity check: rightward_commutation matches stored_expr")

Passed sanity check: rightward_commutation matches stored_expr

In [3]:
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{i, j, k \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, b, c_{1}, c_{2}, \ldots, c_{j}, d_{1}, d_{2}, \ldots, d_{k} \in \mathbb{C}}~\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{j}\cdot d_{1} \cdot  d_{2} \cdot  \ldots \cdot  d_{k}\right) =  \\ \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{j} \cdot b\cdot d_{1} \cdot  d_{2} \cdot  \ldots \cdot  d_{k}\right) \end{array} \end{array}\right)\right]

In [4]:
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 [5]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Operationoperator: 7
operand: 2
1ExprTuple2
2Lambdaparameters: 3
body: 4
3ExprTuple46, 48, 51
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 23
operands: 9
7Literal
8ExprTuple10
9ExprTuple11, 12, 13
10Lambdaparameters: 31
body: 14
11Operationoperator: 54
operands: 15
12Operationoperator: 54
operands: 16
13Operationoperator: 54
operands: 17
14Conditionalvalue: 18
condition: 19
15ExprTuple46, 20
16ExprTuple48, 20
17ExprTuple51, 20
18Operationoperator: 21
operands: 22
19Operationoperator: 23
operands: 24
20Literal
21Literal
22ExprTuple25, 26
23Literal
24ExprTuple27, 28, 29, 30
25Operationoperator: 32
operands: 31
26Operationoperator: 32
operands: 33
27ExprRangelambda_map: 34
start_index: 50
end_index: 46
28Operationoperator: 54
operands: 35
29ExprRangelambda_map: 36
start_index: 50
end_index: 48
30ExprRangelambda_map: 37
start_index: 50
end_index: 51
31ExprTuple38, 42, 39, 40
32Literal
33ExprTuple38, 39, 42, 40
34Lambdaparameter: 64
body: 41
35ExprTuple42, 59
36Lambdaparameter: 64
body: 43
37Lambdaparameter: 64
body: 44
38ExprRangelambda_map: 45
start_index: 50
end_index: 46
39ExprRangelambda_map: 47
start_index: 50
end_index: 48
40ExprRangelambda_map: 49
start_index: 50
end_index: 51
41Operationoperator: 54
operands: 52
42Variable
43Operationoperator: 54
operands: 53
44Operationoperator: 54
operands: 55
45Lambdaparameter: 64
body: 56
46Variable
47Lambdaparameter: 64
body: 57
48Variable
49Lambdaparameter: 64
body: 58
50Literal
51Variable
52ExprTuple56, 59
53ExprTuple57, 59
54Literal
55ExprTuple58, 59
56IndexedVarvariable: 60
index: 64
57IndexedVarvariable: 61
index: 64
58IndexedVarvariable: 62
index: 64
59Literal
60Variable
61Variable
62Variable
63ExprTuple64
64Variable