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 = frac(one, Exp(two, frac(_t, two)))
sub_expr4 = 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, Mult(sub_expr3, sub_expr4)), domain = _m_domain), Sum(index_or_indices = sub_expr1, summand = Mult(sub_expr2, sub_expr3, sub_expr4), 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 \left(\frac{1}{2^{\frac{t}{2}}} \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(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \frac{1}{2^{\frac{t}{2}}} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\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: 60 body: 10
9	Lambda	parameter: 60 body: 12
10	Conditional	value: 13 condition: 15
11	ExprTuple	60
12	Conditional	value: 14 condition: 15
13	Operation	operator: 54 operands: 16
14	Operation	operator: 54 operands: 17
15	Operation	operator: 18 operands: 19
16	ExprTuple	21, 20
17	ExprTuple	21, 27, 28
18	Literal
19	ExprTuple	60, 22
20	Operation	operator: 54 operands: 23
21	Operation	operator: 56 operands: 24
22	Operation	operator: 25 operands: 26
23	ExprTuple	27, 28
24	ExprTuple	38, 29
25	Literal
26	ExprTuple	30, 31
27	Operation	operator: 50 operands: 32
28	Operation	operator: 56 operands: 33
29	Operation	operator: 54 operands: 34
30	Literal
31	Operation	operator: 35 operands: 36
32	ExprTuple	48, 37
33	ExprTuple	38, 39
34	ExprTuple	62, 58, 59, 40, 60
35	Literal
36	ExprTuple	53, 41
37	Operation	operator: 56 operands: 42
38	Literal
39	Operation	operator: 44 operand: 47
40	Literal
41	Operation	operator: 44 operand: 48
42	ExprTuple	62, 46
43	ExprTuple	47
44	Literal
45	ExprTuple	48
46	Operation	operator: 50 operands: 49
47	Operation	operator: 50 operands: 51
48	Literal
49	ExprTuple	63, 62
50	Literal
51	ExprTuple	52, 53
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