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, one, pi, two
from proveit.physics.quantum.QPE import _m_domain, _phase, _t, _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(Mult(Exp(frac(one, Exp(two, frac(_t, two))), two), Sum(index_or_indices = sub_expr1, summand = Mult(sub_expr2, sub_expr3), domain = _m_domain)), Mult(frac(one, _two_pow_t), Sum(index_or_indices = sub_expr1, summand = Mult(sub_expr3, sub_expr2), domain = _m_domain))).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left(\left(\frac{1}{2^{\frac{t}{2}}}\right)^{2} \cdot \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)\right) =  \\ \left(\frac{1}{2^{t}} \cdot \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)\right) \end{array} \end{array}

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.	()	(2)	('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: 60 operands: 5
4	Operation	operator: 60 operands: 6
5	ExprTuple	7, 8
6	ExprTuple	9, 10
7	Operation	operator: 62 operands: 11
8	Operation	operator: 14 operand: 17
9	Operation	operator: 53 operands: 13
10	Operation	operator: 14 operand: 18
11	ExprTuple	16, 68
12	ExprTuple	17
13	ExprTuple	59, 58
14	Literal
15	ExprTuple	18
16	Operation	operator: 53 operands: 19
17	Lambda	parameter: 66 body: 20
18	Lambda	parameter: 66 body: 22
19	ExprTuple	59, 23
20	Conditional	value: 24 condition: 26
21	ExprTuple	66
22	Conditional	value: 25 condition: 26
23	Operation	operator: 62 operands: 27
24	Operation	operator: 60 operands: 28
25	Operation	operator: 60 operands: 29
26	Operation	operator: 30 operands: 31
27	ExprTuple	68, 32
28	ExprTuple	34, 33
29	ExprTuple	33, 34
30	Literal
31	ExprTuple	66, 35
32	Operation	operator: 53 operands: 36
33	Operation	operator: 62 operands: 37
34	Operation	operator: 62 operands: 38
35	Operation	operator: 39 operands: 40
36	ExprTuple	69, 68
37	ExprTuple	42, 41
38	ExprTuple	42, 43
39	Literal
40	ExprTuple	44, 45
41	Operation	operator: 55 operand: 50
42	Literal
43	Operation	operator: 60 operands: 47
44	Literal
45	Operation	operator: 48 operands: 49
46	ExprTuple	50
47	ExprTuple	68, 64, 65, 51, 66
48	Literal
49	ExprTuple	58, 52
50	Operation	operator: 53 operands: 54
51	Literal
52	Operation	operator: 55 operand: 59
53	Literal
54	ExprTuple	57, 58
55	Literal
56	ExprTuple	59
57	Operation	operator: 60 operands: 61
58	Operation	operator: 62 operands: 63
59	Literal
60	Literal
61	ExprTuple	68, 64, 65, 66, 67
62	Literal
63	ExprTuple	68, 69
64	Literal
65	Literal
66	Variable
67	Variable
68	Literal
69	Literal