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, 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
expr = Equals(_alpha_m_mod_two_pow_t, Mult(frac(one, _two_pow_t), Sum(index_or_indices = [k], 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())

\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)

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: 33 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Operation	operator: 11 operands: 50
9	Operation	operator: 49 operands: 12
10	Operation	operator: 13 operand: 15
11	Literal
12	ExprTuple	45, 52
13	Literal
14	ExprTuple	15
15	Lambda	parameter: 24 body: 17
16	ExprTuple	24
17	Conditional	value: 18 condition: 19
18	Operation	operator: 53 operands: 20
19	Operation	operator: 21 operands: 22
20	ExprTuple	23, 24
21	Literal
22	ExprTuple	24, 25
23	Operation	operator: 53 operands: 26
24	Variable
25	Operation	operator: 27 operands: 28
26	ExprTuple	29, 30
27	Literal
28	ExprTuple	31, 32
29	Literal
30	Operation	operator: 33 operands: 34
31	Literal
32	Operation	operator: 40 operands: 35
33	Literal
34	ExprTuple	55, 36, 37, 38
35	ExprTuple	52, 39
36	Literal
37	Literal
38	Operation	operator: 40 operands: 41
39	Operation	operator: 46 operand: 45
40	Literal
41	ExprTuple	43, 44
42	ExprTuple	45
43	Literal
44	Operation	operator: 46 operand: 48
45	Literal
46	Literal
47	ExprTuple	48
48	Operation	operator: 49 operands: 50
49	Literal
50	ExprTuple	51, 52
51	Variable
52	Operation	operator: 53 operands: 54
53	Literal
54	ExprTuple	55, 56
55	Literal
56	Literal