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, subtract, two
from proveit.physics.quantum.QPE import _alpha_m_mod_two_pow_t, _m_domain, _phase, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = frac(one, _two_pow_t)
expr = Equals(Equals(_alpha_m_mod_two_pow_t, Mult(sub_expr2, Sum(index_or_indices = sub_expr1, summand = Mult(Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t))), Exp(e, Mult(two, pi, i, _phase, k))), domain = _m_domain))), Equals(_alpha_m_mod_two_pow_t, Mult(sub_expr2, Sum(index_or_indices = sub_expr1, summand = Exp(Exp(e, Mult(two, pi, i, subtract(_phase, frac(m, _two_pow_t)))), k), 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(\alpha_{m ~\textup{mod}~ 2^{t}} = \left(\frac{1}{2^{t}} \cdot \left(\sum_{k = 0}^{2^{t} - 1} \left(\mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}} \cdot \mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k}\right)\right)\right)\right) = \left(\alpha_{m ~\textup{mod}~ 2^{t}} = \left(\frac{1}{2^{t}} \cdot \left(\sum_{k = 0}^{2^{t} - 1} (\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \left(\varphi - \frac{m}{2^{t}}\right)})^{k}\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: 5 operands: 1
1	ExprTuple	2, 3
2	Operation	operator: 5 operands: 4
3	Operation	operator: 5 operands: 6
4	ExprTuple	8, 7
5	Literal
6	ExprTuple	8, 9
7	Operation	operator: 65 operands: 10
8	Operation	operator: 11 operand: 15
9	Operation	operator: 65 operands: 13
10	ExprTuple	16, 14
11	Literal
12	ExprTuple	15
13	ExprTuple	16, 17
14	Operation	operator: 21 operand: 23
15	Operation	operator: 19 operands: 74
16	Operation	operator: 73 operands: 20
17	Operation	operator: 21 operand: 24
18	ExprTuple	23
19	Literal
20	ExprTuple	64, 76
21	Literal
22	ExprTuple	24
23	Lambda	parameter: 71 body: 25
24	Lambda	parameter: 71 body: 27
25	Conditional	value: 28 condition: 30
26	ExprTuple	71
27	Conditional	value: 29 condition: 30
28	Operation	operator: 65 operands: 31
29	Operation	operator: 77 operands: 32
30	Operation	operator: 33 operands: 34
31	ExprTuple	35, 36
32	ExprTuple	37, 71
33	Literal
34	ExprTuple	71, 38
35	Operation	operator: 77 operands: 39
36	Operation	operator: 77 operands: 40
37	Operation	operator: 77 operands: 41
38	Operation	operator: 42 operands: 43
39	ExprTuple	46, 44
40	ExprTuple	46, 45
41	ExprTuple	46, 47
42	Literal
43	ExprTuple	48, 49
44	Operation	operator: 67 operand: 54
45	Operation	operator: 65 operands: 51
46	Literal
47	Operation	operator: 65 operands: 52
48	Literal
49	Operation	operator: 58 operands: 53
50	ExprTuple	54
51	ExprTuple	79, 69, 70, 62, 71
52	ExprTuple	79, 69, 70, 55
53	ExprTuple	76, 56
54	Operation	operator: 73 operands: 57
55	Operation	operator: 58 operands: 59
56	Operation	operator: 67 operand: 64
57	ExprTuple	61, 76
58	Literal
59	ExprTuple	62, 63
60	ExprTuple	64
61	Operation	operator: 65 operands: 66
62	Literal
63	Operation	operator: 67 operand: 72
64	Literal
65	Literal
66	ExprTuple	79, 69, 70, 71, 75
67	Literal
68	ExprTuple	72
69	Literal
70	Literal
71	Variable
72	Operation	operator: 73 operands: 74
73	Literal
74	ExprTuple	75, 76
75	Variable
76	Operation	operator: 77 operands: 78
77	Literal
78	ExprTuple	79, 80
79	Literal
80	Literal