# 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 i, j, k
from proveit.core_expr_types import a_1_to_i, b_1_to_j, c_1_to_k
from proveit.logic import Equals, Forall
from proveit.numbers import Complex, Mult, Natural

In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i, j, k], instance_expr = Forall(instance_param_or_params = [a_1_to_i, b_1_to_j, c_1_to_k], instance_expr = Equals(Mult(a_1_to_i, Mult(b_1_to_j), c_1_to_k), Mult(a_1_to_i, b_1_to_j, c_1_to_k)).with_wrapping_at(2), domain = Complex), 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_{i, j, k \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, b_{1}, b_{2}, \ldots, b_{j}, c_{1}, c_{2}, \ldots, c_{k} \in \mathbb{C}}~\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot \left(b_{1} \cdot  b_{2} \cdot  \ldots \cdot  b_{j}\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) =  \\ \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i}\cdot b_{1} \cdot  b_{2} \cdot  \ldots \cdot  b_{j}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) \end{array} \end{array}\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
2Lambdaparameters: 3
body: 4
3ExprTuple44, 57, 46
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 23
operands: 9
7Literal
8ExprTuple10
9ExprTuple11, 12, 13
10Lambdaparameters: 31
body: 14
11Operationoperator: 49
operands: 15
12Operationoperator: 49
operands: 16
13Operationoperator: 49
operands: 17
14Conditionalvalue: 18
condition: 19
15ExprTuple44, 20
16ExprTuple57, 20
17ExprTuple46, 20
18Operationoperator: 21
operands: 22
19Operationoperator: 23
operands: 24
20Literal
21Literal
22ExprTuple25, 26
23Literal
24ExprTuple27, 28, 29
25Operationoperator: 41
operands: 30
26Operationoperator: 41
operands: 31
27ExprRangelambda_map: 32
start_index: 56
end_index: 44
28ExprRangelambda_map: 33
start_index: 56
end_index: 57
29ExprRangelambda_map: 34
start_index: 56
end_index: 46
30ExprTuple36, 35, 37
31ExprTuple36, 51, 37
32Lambdaparameter: 63
body: 38
33Lambdaparameter: 63
body: 39
34Lambdaparameter: 63
body: 40
35Operationoperator: 41
operands: 42
36ExprRangelambda_map: 43
start_index: 56
end_index: 44
37ExprRangelambda_map: 45
start_index: 56
end_index: 46
38Operationoperator: 49
operands: 47
39Operationoperator: 49
operands: 48
40Operationoperator: 49
operands: 50
41Literal
42ExprTuple51
43Lambdaparameter: 63
body: 52
44Variable
45Lambdaparameter: 63
body: 53
46Variable
47ExprTuple52, 54
48ExprTuple60, 54
49Literal
50ExprTuple53, 54
51ExprRangelambda_map: 55
start_index: 56
end_index: 57
52IndexedVarvariable: 58
index: 63
53IndexedVarvariable: 59
index: 63
54Literal
55Lambdaparameter: 63
body: 60
56Literal
57Variable
58Variable
59Variable
60IndexedVarvariable: 61
index: 63
61Variable
62ExprTuple63
63Variable