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
expr = Equals(_alpha_m_mod_two_pow_t, Mult(frac(one, _two_pow_t), Sum(index_or_indices = [k], summand = Mult(Exp(e, Neg(frac(Mult(two, pi, i, k, Mod(m, _two_pow_t)), _two_pow_t))), Exp(e, Mult(two, pi, i, _phase, 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} \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)

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: 51
4	Operation	operator: 46 operands: 7
5	Literal
6	ExprTuple	51
7	ExprTuple	8, 9
8	Operation	operator: 40 operands: 10
9	Operation	operator: 11 operand: 13
10	ExprTuple	45, 55
11	Literal
12	ExprTuple	13
13	Lambda	parameter: 50 body: 15
14	ExprTuple	50
15	Conditional	value: 16 condition: 17
16	Operation	operator: 46 operands: 18
17	Operation	operator: 19 operands: 20
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	50, 23
21	Operation	operator: 56 operands: 24
22	Operation	operator: 56 operands: 25
23	Operation	operator: 26 operands: 27
24	ExprTuple	29, 28
25	ExprTuple	29, 30
26	Literal
27	ExprTuple	31, 32
28	Operation	operator: 42 operand: 37
29	Literal
30	Operation	operator: 46 operands: 34
31	Literal
32	Operation	operator: 35 operands: 36
33	ExprTuple	37
34	ExprTuple	58, 48, 49, 38, 50
35	Literal
36	ExprTuple	55, 39
37	Operation	operator: 40 operands: 41
38	Literal
39	Operation	operator: 42 operand: 45
40	Literal
41	ExprTuple	44, 55
42	Literal
43	ExprTuple	45
44	Operation	operator: 46 operands: 47
45	Literal
46	Literal
47	ExprTuple	58, 48, 49, 50, 51
48	Literal
49	Literal
50	Variable
51	Operation	operator: 52 operands: 53
52	Literal
53	ExprTuple	54, 55
54	Variable
55	Operation	operator: 56 operands: 57
56	Literal
57	ExprTuple	58, 59
58	Literal
59	Literal