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

# build up the expression from sub-expressions
sub_expr1 = Sin(Mult(pi, Abs(subtract(_delta_b_floor, frac(l, _two_pow_t)))))
sub_expr2 = Abs(subtract(one, Exp(e, Mult(two, pi, i, subtract(Mult(_two_pow_t, _delta_b_floor), l)))))
expr = Equals(Mult(frac(one, _two_pow_t), frac(sub_expr2, Mult(two, sub_expr1))), frac(sub_expr2, Mult(two, _two_pow_t, sub_expr1)))

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