# 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 Conditional, Lambda, i, j
from proveit.core_expr_types import a_1_to_i, b_1_to_j
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import Add, Complex, Natural, zero

In [2]:
# build up the expression from sub-expressions
expr = Lambda([i, j], Conditional(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), And(InSet(i, Natural), InSet(j, 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())

\left(i, j\right) \mapsto \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) \textrm{ if } i \in \mathbb{N} ,  j \in \mathbb{N}\right..

In [5]:
stored_expr.style_options()

no style options
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Lambdaparameters: 1
body: 2
1ExprTuple36, 39
2Conditionalvalue: 3
condition: 4
3Operationoperator: 5
operand: 8
4Operationoperator: 19
operands: 7
5Literal
6ExprTuple8
7ExprTuple9, 10
8Lambdaparameters: 27
body: 11
9Operationoperator: 41
operands: 12
10Operationoperator: 41
operands: 13
11Conditionalvalue: 14
condition: 15
12ExprTuple36, 16
13ExprTuple39, 16
14Operationoperator: 17
operands: 18
15Operationoperator: 19
operands: 20
16Literal
17Literal
18ExprTuple21, 22
19Literal
20ExprTuple23, 24
21Operationoperator: 26
operands: 25
22Operationoperator: 26
operands: 27
23ExprRangelambda_map: 28
start_index: 38
end_index: 36
24ExprRangelambda_map: 29
start_index: 38
end_index: 39
25ExprTuple31, 30, 32
26Literal
27ExprTuple31, 32
28Lambdaparameter: 49
body: 33
29Lambdaparameter: 49
body: 34
30Literal
31ExprRangelambda_map: 35
start_index: 38
end_index: 36
32ExprRangelambda_map: 37
start_index: 38
end_index: 39
33Operationoperator: 41
operands: 40
34Operationoperator: 41
operands: 42
35Lambdaparameter: 49
body: 43
36Variable
37Lambdaparameter: 49
body: 44
38Literal
39Variable
40ExprTuple43, 45
41Literal
42ExprTuple44, 45
43IndexedVarvariable: 46
index: 49
44IndexedVarvariable: 47
index: 49
45Literal
46Variable
47Variable
48ExprTuple49
49Variable