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 = frac(one, Exp(two, frac(_t, two)))
sub_expr3 = Mult(Exp(e, Mult(two, pi, i, _phase, k)), Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t))))
expr = Equals(Mult(sub_expr2, Sum(index_or_indices = sub_expr1, summand = sub_expr3, domain = _m_domain)), Sum(index_or_indices = sub_expr1, summand = Mult(sub_expr2, sub_expr3), 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(\frac{1}{2^{\frac{t}{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(\sum_{k = 0}^{2^{t} - 1} \left(\frac{1}{2^{\frac{t}{2}}} \cdot \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) \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: 54 operands: 5
4	Operation	operator: 9 operand: 8
5	ExprTuple	18, 7
6	ExprTuple	8
7	Operation	operator: 9 operand: 12
8	Lambda	parameter: 60 body: 11
9	Literal
10	ExprTuple	12
11	Conditional	value: 13 condition: 17
12	Lambda	parameter: 60 body: 15
13	Operation	operator: 54 operands: 16
14	ExprTuple	60
15	Conditional	value: 19 condition: 17
16	ExprTuple	18, 19
17	Operation	operator: 20 operands: 21
18	Operation	operator: 49 operands: 22
19	Operation	operator: 54 operands: 23
20	Literal
21	ExprTuple	60, 24
22	ExprTuple	51, 25
23	ExprTuple	26, 27
24	Operation	operator: 28 operands: 29
25	Operation	operator: 56 operands: 30
26	Operation	operator: 56 operands: 31
27	Operation	operator: 56 operands: 32
28	Literal
29	ExprTuple	33, 34
30	ExprTuple	62, 35
31	ExprTuple	37, 36
32	ExprTuple	37, 38
33	Literal
34	Operation	operator: 39 operands: 40
35	Operation	operator: 49 operands: 41
36	Operation	operator: 54 operands: 42
37	Literal
38	Operation	operator: 47 operand: 46
39	Literal
40	ExprTuple	53, 44
41	ExprTuple	63, 62
42	ExprTuple	62, 58, 59, 45, 60
43	ExprTuple	46
44	Operation	operator: 47 operand: 51
45	Literal
46	Operation	operator: 49 operands: 50
47	Literal
48	ExprTuple	51
49	Literal
50	ExprTuple	52, 53
51	Literal
52	Operation	operator: 54 operands: 55
53	Operation	operator: 56 operands: 57
54	Literal
55	ExprTuple	62, 58, 59, 60, 61
56	Literal
57	ExprTuple	62, 63
58	Literal
59	Literal
60	Variable
61	Variable
62	Literal
63	Literal