Expression of type Equals

from the theory of proveit.linear_algebra.addition

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 Function, N, k, v, vk
from proveit.linear_algebra import ScalarMult, VecAdd, VecSum
from proveit.logic import Equals
from proveit.numbers import Exp, Interval, Mult, e, i, one, pi, two, zero

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = ScalarMult(Exp(e, Mult(two, pi, i, k)), vk)
expr = Equals(VecSum(index_or_indices = sub_expr1, summand = sub_expr2, domain = Interval(zero, N)), VecAdd(Function(v, [zero]), VecSum(index_or_indices = sub_expr1, summand = sub_expr2, domain = Interval(one, N))))

expr:

# 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

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left(\sum_{k=0}^{N} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot k} \cdot v\left(k\right)\right)\right) = \left(v\left(0\right) + \left(\sum_{k=1}^{N} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot k} \cdot v\left(k\right)\right)\right)\right)

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 13 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	9, 10
8	Lambda	parameter: 46 body: 11
9	Operation	operator: 33 operand: 27
10	Operation	operator: 13 operand: 16
11	Conditional	value: 20 condition: 15
12	ExprTuple	27
13	Literal
14	ExprTuple	16
15	Operation	operator: 25 operands: 17
16	Lambda	parameter: 46 body: 18
17	ExprTuple	46, 19
18	Conditional	value: 20 condition: 21
19	Operation	operator: 35 operands: 22
20	Operation	operator: 23 operands: 24
21	Operation	operator: 25 operands: 26
22	ExprTuple	27, 40
23	Literal
24	ExprTuple	28, 29
25	Literal
26	ExprTuple	46, 30
27	Literal
28	Operation	operator: 31 operands: 32
29	Operation	operator: 33 operand: 46
30	Operation	operator: 35 operands: 36
31	Literal
32	ExprTuple	37, 38
33	Variable
34	ExprTuple	46
35	Literal
36	ExprTuple	39, 40
37	Literal
38	Operation	operator: 41 operands: 42
39	Literal
40	Variable
41	Literal
42	ExprTuple	43, 44, 45, 46
43	Literal
44	Literal
45	Literal
46	Variable