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, l
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, Sum, e, frac, i, one, pi, subtract, two
from proveit.physics.quantum.QPE import _delta_b_floor, _m_domain, _two_pow_t

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

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} (\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(\delta_{b_{\textit{f}}} - \frac{l}{2^{t}}\right)})^{k}\right) = \frac{1 - \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(\delta_{b_{\textit{f}}} - \frac{l}{2^{t}}\right) \cdot 2^{t}}}{1 - \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(\delta_{b_{\textit{f}}} - \frac{l}{2^{t}}\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: 5 operand: 8
4	Operation	operator: 57 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Lambda	parameter: 24 body: 12
9	Operation	operator: 46 operands: 13
10	Operation	operator: 46 operands: 14
11	ExprTuple	24
12	Conditional	value: 15 condition: 16
13	ExprTuple	48, 17
14	ExprTuple	48, 18
15	Operation	operator: 61 operands: 19
16	Operation	operator: 20 operands: 21
17	Operation	operator: 53 operand: 26
18	Operation	operator: 53 operand: 27
19	ExprTuple	27, 24
20	Literal
21	ExprTuple	24, 25
22	ExprTuple	26
23	ExprTuple	27
24	Variable
25	Operation	operator: 28 operands: 29
26	Operation	operator: 61 operands: 30
27	Operation	operator: 61 operands: 31
28	Literal
29	ExprTuple	32, 33
30	ExprTuple	35, 34
31	ExprTuple	35, 36
32	Literal
33	Operation	operator: 46 operands: 37
34	Operation	operator: 39 operands: 38
35	Literal
36	Operation	operator: 39 operands: 40
37	ExprTuple	60, 41
38	ExprTuple	63, 42, 43, 44, 60
39	Literal
40	ExprTuple	63, 42, 43, 44
41	Operation	operator: 53 operand: 48
42	Literal
43	Literal
44	Operation	operator: 46 operands: 47
45	ExprTuple	48
46	Literal
47	ExprTuple	49, 50
48	Literal
49	Operation	operator: 51 operand: 55
50	Operation	operator: 53 operand: 56
51	Literal
52	ExprTuple	55
53	Literal
54	ExprTuple	56
55	Literal
56	Operation	operator: 57 operands: 58
57	Literal
58	ExprTuple	59, 60
59	Variable
60	Operation	operator: 61 operands: 62
61	Literal
62	ExprTuple	63, 64
63	Literal
64	Literal