# 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.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Conditional, Lambda, c, i, j, k
from proveit.core_expr_types import a_1_to_i, b_1_to_j, d_1_to_k
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import Add, Complex, Natural

In [2]:
# build up the expression from sub-expressions
expr = Lambda([i, j, k], Conditional(Forall(instance_param_or_params = [a_1_to_i, b_1_to_j, c, d_1_to_k], instance_expr = Equals(Add(a_1_to_i, b_1_to_j, c, d_1_to_k), Add(a_1_to_i, c, b_1_to_j, d_1_to_k)).with_wrapping_at(2), domain = Complex), And(InSet(i, Natural), InSet(j, Natural), InSet(k, 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, k\right) \mapsto \left\{\forall_{a_{1}, a_{2}, \ldots, a_{i}, b_{1}, b_{2}, \ldots, b_{j}, c, d_{1}, d_{2}, \ldots, d_{k} \in \mathbb{C}}~\left(\begin{array}{c} \begin{array}{l} \left(a_{1} +  a_{2} +  \ldots +  a_{i}+ b_{1} +  b_{2} +  \ldots +  b_{j} + c+ d_{1} +  d_{2} +  \ldots +  d_{k}\right) =  \\ \left(a_{1} +  a_{2} +  \ldots +  a_{i} + c+ b_{1} +  b_{2} +  \ldots +  b_{j}+ d_{1} +  d_{2} +  \ldots +  d_{k}\right) \end{array} \end{array}\right) \textrm{ if } i \in \mathbb{N} ,  j \in \mathbb{N} ,  k \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
1ExprTuple44, 46, 49
2Conditionalvalue: 3
condition: 4
3Operationoperator: 5
operand: 8
4Operationoperator: 21
operands: 7
5Literal
6ExprTuple8
7ExprTuple9, 10, 11
8Lambdaparameters: 29
body: 12
9Operationoperator: 52
operands: 13
10Operationoperator: 52
operands: 14
11Operationoperator: 52
operands: 15
12Conditionalvalue: 16
condition: 17
13ExprTuple44, 18
14ExprTuple46, 18
15ExprTuple49, 18
16Operationoperator: 19
operands: 20
17Operationoperator: 21
operands: 22
18Literal
19Literal
20ExprTuple23, 24
21Literal
22ExprTuple25, 26, 27, 28
23Operationoperator: 30
operands: 29
24Operationoperator: 30
operands: 31
25ExprRangelambda_map: 32
start_index: 48
end_index: 44
26ExprRangelambda_map: 33
start_index: 48
end_index: 46
27Operationoperator: 52
operands: 34
28ExprRangelambda_map: 35
start_index: 48
end_index: 49
29ExprTuple36, 37, 41, 38
30Literal
31ExprTuple36, 41, 37, 38
32Lambdaparameter: 62
body: 39
33Lambdaparameter: 62
body: 40
34ExprTuple41, 57
35Lambdaparameter: 62
body: 42
36ExprRangelambda_map: 43
start_index: 48
end_index: 44
37ExprRangelambda_map: 45
start_index: 48
end_index: 46
38ExprRangelambda_map: 47
start_index: 48
end_index: 49
39Operationoperator: 52
operands: 50
40Operationoperator: 52
operands: 51
41Variable
42Operationoperator: 52
operands: 53
43Lambdaparameter: 62
body: 54
44Variable
45Lambdaparameter: 62
body: 55
46Variable
47Lambdaparameter: 62
body: 56
48Literal
49Variable
50ExprTuple54, 57
51ExprTuple55, 57
52Literal
53ExprTuple56, 57
54IndexedVarvariable: 58
index: 62
55IndexedVarvariable: 59
index: 62
56IndexedVarvariable: 60
index: 62
57Literal
58Variable
59Variable
60Variable
61ExprTuple62
62Variable