# 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, ExprTuple, 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 = ExprTuple(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(\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..\right)

In [5]:
stored_expr.style_options()

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

core typesub-expressionsexpression
0ExprTuple1
1Lambdaparameters: 2
body: 3
2ExprTuple45, 47, 50
3Conditionalvalue: 4
condition: 5
4Operationoperator: 6
operand: 9
5Operationoperator: 22
operands: 8
6Literal
7ExprTuple9
8ExprTuple10, 11, 12
9Lambdaparameters: 30
body: 13
10Operationoperator: 53
operands: 14
11Operationoperator: 53
operands: 15
12Operationoperator: 53
operands: 16
13Conditionalvalue: 17
condition: 18
14ExprTuple45, 19
15ExprTuple47, 19
16ExprTuple50, 19
17Operationoperator: 20
operands: 21
18Operationoperator: 22
operands: 23
19Literal
20Literal
21ExprTuple24, 25
22Literal
23ExprTuple26, 27, 28, 29
24Operationoperator: 31
operands: 30
25Operationoperator: 31
operands: 32
26ExprRangelambda_map: 33
start_index: 49
end_index: 45
27ExprRangelambda_map: 34
start_index: 49
end_index: 47
28Operationoperator: 53
operands: 35
29ExprRangelambda_map: 36
start_index: 49
end_index: 50
30ExprTuple37, 38, 42, 39
31Literal
32ExprTuple37, 42, 38, 39
33Lambdaparameter: 63
body: 40
34Lambdaparameter: 63
body: 41
35ExprTuple42, 58
36Lambdaparameter: 63
body: 43
37ExprRangelambda_map: 44
start_index: 49
end_index: 45
38ExprRangelambda_map: 46
start_index: 49
end_index: 47
39ExprRangelambda_map: 48
start_index: 49
end_index: 50
40Operationoperator: 53
operands: 51
41Operationoperator: 53
operands: 52
42Variable
43Operationoperator: 53
operands: 54
44Lambdaparameter: 63
body: 55
45Variable
46Lambdaparameter: 63
body: 56
47Variable
48Lambdaparameter: 63
body: 57
49Literal
50Variable
51ExprTuple55, 58
52ExprTuple56, 58
53Literal
54ExprTuple57, 58
55IndexedVarvariable: 59
index: 63
56IndexedVarvariable: 60
index: 63
57IndexedVarvariable: 61
index: 63
58Literal
59Variable
60Variable
61Variable
62ExprTuple63
63Variable