# from the theory of proveit.numbers.addition¶

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
from proveit.core_expr_types import a_1_to_i, b_1_to_j
from proveit.logic import Equals, Forall
from proveit.numbers import Add, Complex, Natural, zero

In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i, j], instance_expr = Forall(instance_param_or_params = [a_1_to_i, b_1_to_j], instance_expr = Equals(Add(a_1_to_i, zero, b_1_to_j), Add(a_1_to_i, b_1_to_j)), 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 \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, b_{1}, b_{2}, \ldots, b_{j} \in \mathbb{C}}~\left(\left(a_{1} +  a_{2} +  \ldots +  a_{i} + 0+ b_{1} +  b_{2} +  \ldots +  b_{j}\right) = \left(a_{1} +  a_{2} +  \ldots +  a_{i}+ b_{1} +  b_{2} +  \ldots +  b_{j}\right)\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
3ExprTuple38, 41
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 21
operands: 9
7Literal
8ExprTuple10
9ExprTuple11, 12
10Lambdaparameters: 29
body: 13
11Operationoperator: 43
operands: 14
12Operationoperator: 43
operands: 15
13Conditionalvalue: 16
condition: 17
14ExprTuple38, 18
15ExprTuple41, 18
16Operationoperator: 19
operands: 20
17Operationoperator: 21
operands: 22
18Literal
19Literal
20ExprTuple23, 24
21Literal
22ExprTuple25, 26
23Operationoperator: 28
operands: 27
24Operationoperator: 28
operands: 29
25ExprRangelambda_map: 30
start_index: 40
end_index: 38
26ExprRangelambda_map: 31
start_index: 40
end_index: 41
27ExprTuple33, 32, 34
28Literal
29ExprTuple33, 34
30Lambdaparameter: 51
body: 35
31Lambdaparameter: 51
body: 36
32Literal
33ExprRangelambda_map: 37
start_index: 40
end_index: 38
34ExprRangelambda_map: 39
start_index: 40
end_index: 41
35Operationoperator: 43
operands: 42
36Operationoperator: 43
operands: 44
37Lambdaparameter: 51
body: 45
38Variable
39Lambdaparameter: 51
body: 46
40Literal
41Variable
42ExprTuple45, 47
43Literal
44ExprTuple46, 47
45IndexedVarvariable: 48
index: 51
46IndexedVarvariable: 49
index: 51
47Literal
48Variable
49Variable
50ExprTuple51
51Variable