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 = frac(one, _two_pow_t)
sub_expr2 = subtract(_delta_b_floor, frac(l, _two_pow_t))
sub_expr3 = subtract(one, Exp(e, Mult(two, pi, i, subtract(Mult(_two_pow_t, _delta_b_floor), l))))
expr = Equals(Abs(Mult(sub_expr1, frac(sub_expr3, subtract(one, Exp(e, Mult(two, pi, i, sub_expr2)))))), Mult(sub_expr1, frac(Abs(sub_expr3), Mult(two, Sin(Mult(pi, Abs(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|\frac{1}{2^{t}} \cdot \frac{1 - \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(\left(2^{t} \cdot \delta_{b_{\textit{f}}}\right) - l\right)}}{1 - \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(\delta_{b_{\textit{f}}} - \frac{l}{2^{t}}\right)}}\right| = \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)

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: 38 operand: 7
4	Operation	operator: 56 operands: 6
5	ExprTuple	7
6	ExprTuple	11, 8
7	Operation	operator: 56 operands: 9
8	Operation	operator: 62 operands: 10
9	ExprTuple	11, 12
10	ExprTuple	13, 14
11	Operation	operator: 62 operands: 15
12	Operation	operator: 62 operands: 16
13	Operation	operator: 38 operand: 20
14	Operation	operator: 56 operands: 18
15	ExprTuple	27, 67
16	ExprTuple	20, 19
17	ExprTuple	20
18	ExprTuple	71, 21
19	Operation	operator: 50 operands: 22
20	Operation	operator: 50 operands: 23
21	Operation	operator: 24 operand: 29
22	ExprTuple	27, 26
23	ExprTuple	27, 28
24	Literal
25	ExprTuple	29
26	Operation	operator: 58 operand: 33
27	Literal
28	Operation	operator: 58 operand: 34
29	Operation	operator: 56 operands: 32
30	ExprTuple	33
31	ExprTuple	34
32	ExprTuple	46, 35
33	Operation	operator: 69 operands: 36
34	Operation	operator: 69 operands: 37
35	Operation	operator: 38 operand: 45
36	ExprTuple	41, 40
37	ExprTuple	41, 42
38	Literal
39	ExprTuple	45
40	Operation	operator: 56 operands: 43
41	Literal
42	Operation	operator: 56 operands: 44
43	ExprTuple	71, 46, 47, 45
44	ExprTuple	71, 46, 47, 48
45	Operation	operator: 50 operands: 49
46	Literal
47	Literal
48	Operation	operator: 50 operands: 51
49	ExprTuple	61, 52
50	Literal
51	ExprTuple	53, 54
52	Operation	operator: 58 operand: 60
53	Operation	operator: 56 operands: 57
54	Operation	operator: 58 operand: 66
55	ExprTuple	60
56	Literal
57	ExprTuple	67, 61
58	Literal
59	ExprTuple	66
60	Operation	operator: 62 operands: 63
61	Operation	operator: 64 operand: 68
62	Literal
63	ExprTuple	66, 67
64	Literal
65	ExprTuple	68
66	Variable
67	Operation	operator: 69 operands: 70
68	Literal
69	Literal
70	ExprTuple	71, 72
71	Literal
72	Literal