Expression of type Equals

from the theory of proveit.physics.quantum.QPE

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 k, m
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, Neg, Sum, e, frac, i, pi, two
from proveit.physics.quantum.QPE import _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr3 = Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t)))
expr = Equals(Sum(index_or_indices = sub_expr1, summand = Mult(sub_expr2, sub_expr3), domain = _m_domain), Sum(index_or_indices = sub_expr1, summand = Mult(sub_expr3, sub_expr2), domain = _m_domain))

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}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right)\right) = \left(\sum_{k = 0}^{2^{t} - 1} \left(\mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k}\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: 6 operand: 8
4	Operation	operator: 6 operand: 9
5	ExprTuple	8
6	Literal
7	ExprTuple	9
8	Lambda	parameter: 52 body: 10
9	Lambda	parameter: 52 body: 12
10	Conditional	value: 13 condition: 15
11	ExprTuple	52
12	Conditional	value: 14 condition: 15
13	Operation	operator: 46 operands: 16
14	Operation	operator: 46 operands: 17
15	Operation	operator: 18 operands: 19
16	ExprTuple	21, 20
17	ExprTuple	20, 21
18	Literal
19	ExprTuple	52, 22
20	Operation	operator: 48 operands: 23
21	Operation	operator: 48 operands: 24
22	Operation	operator: 25 operands: 26
23	ExprTuple	28, 27
24	ExprTuple	28, 29
25	Literal
26	ExprTuple	30, 31
27	Operation	operator: 41 operand: 36
28	Literal
29	Operation	operator: 46 operands: 33
30	Literal
31	Operation	operator: 34 operands: 35
32	ExprTuple	36
33	ExprTuple	54, 50, 51, 37, 52
34	Literal
35	ExprTuple	44, 38
36	Operation	operator: 39 operands: 40
37	Literal
38	Operation	operator: 41 operand: 45
39	Literal
40	ExprTuple	43, 44
41	Literal
42	ExprTuple	45
43	Operation	operator: 46 operands: 47
44	Operation	operator: 48 operands: 49
45	Literal
46	Literal
47	ExprTuple	54, 50, 51, 52, 53
48	Literal
49	ExprTuple	54, 55
50	Literal
51	Literal
52	Variable
53	Variable
54	Literal
55	Literal