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, Mod, Mult, Neg, Sum, e, frac, i, one, pi, 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)
sub_expr3 = Exp(e, Mult(two, pi, i, _phase, k))
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, Mod(m, _two_pow_t)), _two_pow_t))), sub_expr3), domain = _m_domain))), 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))), sub_expr3), 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 \left(m ~\textup{mod}~ 2^{t}\right)}{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} \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)

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: 66 operands: 10
8	Operation	operator: 11 operand: 68
9	Operation	operator: 66 operands: 13
10	ExprTuple	15, 14
11	Literal
12	ExprTuple	68
13	ExprTuple	15, 16
14	Operation	operator: 19 operand: 21
15	Operation	operator: 58 operands: 18
16	Operation	operator: 19 operand: 22
17	ExprTuple	21
18	ExprTuple	64, 75
19	Literal
20	ExprTuple	22
21	Lambda	parameter: 71 body: 23
22	Lambda	parameter: 71 body: 25
23	Conditional	value: 26 condition: 28
24	ExprTuple	71
25	Conditional	value: 27 condition: 28
26	Operation	operator: 66 operands: 29
27	Operation	operator: 66 operands: 30
28	Operation	operator: 31 operands: 32
29	ExprTuple	33, 35
30	ExprTuple	34, 35
31	Literal
32	ExprTuple	71, 36
33	Operation	operator: 76 operands: 37
34	Operation	operator: 76 operands: 38
35	Operation	operator: 76 operands: 39
36	Operation	operator: 40 operands: 41
37	ExprTuple	44, 42
38	ExprTuple	44, 43
39	ExprTuple	44, 45
40	Literal
41	ExprTuple	46, 47
42	Operation	operator: 60 operand: 53
43	Operation	operator: 60 operand: 54
44	Literal
45	Operation	operator: 66 operands: 50
46	Literal
47	Operation	operator: 51 operands: 52
48	ExprTuple	53
49	ExprTuple	54
50	ExprTuple	78, 69, 70, 55, 71
51	Literal
52	ExprTuple	75, 56
53	Operation	operator: 58 operands: 57
54	Operation	operator: 58 operands: 59
55	Literal
56	Operation	operator: 60 operand: 64
57	ExprTuple	62, 75
58	Literal
59	ExprTuple	63, 75
60	Literal
61	ExprTuple	64
62	Operation	operator: 66 operands: 65
63	Operation	operator: 66 operands: 67
64	Literal
65	ExprTuple	78, 69, 70, 71, 68
66	Literal
67	ExprTuple	78, 69, 70, 71, 74
68	Operation	operator: 72 operands: 73
69	Literal
70	Literal
71	Variable
72	Literal
73	ExprTuple	74, 75
74	Variable
75	Operation	operator: 76 operands: 77
76	Literal
77	ExprTuple	78, 79
78	Literal
79	Literal