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 Function, Variable, k
from proveit.logic import Equals
from proveit.numbers import Exp, Mult, Sum, e, frac, i, one, pi, two
from proveit.physics.quantum.QPE import _alpha_m, _m_domain, _phase, _t

# build up the expression from sub-expressions
sub_expr1 = [k]
expr = Equals(_alpha_m, Mult(frac(one, Exp(two, frac(_t, two))), Sum(index_or_indices = sub_expr1, summand = Mult(Exp(e, Mult(two, pi, i, _phase, k)), Function(Variable("_a", latex_format = r"{_{-}a}"), sub_expr1)), 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} = \left(\frac{1}{2^{\frac{t}{2}}} \cdot \left(\sum_{k = 0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot {_{-}a}\left(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: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 38 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 10
8	Variable
9	Operation	operator: 21 operands: 11
10	Operation	operator: 12 operand: 15
11	ExprTuple	54, 14
12	Literal
13	ExprTuple	15
14	Operation	operator: 48 operands: 16
15	Lambda	parameter: 45 body: 17
16	ExprTuple	52, 18
17	Conditional	value: 19 condition: 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 38 operands: 23
20	Operation	operator: 24 operands: 25
21	Literal
22	ExprTuple	53, 52
23	ExprTuple	26, 27
24	Literal
25	ExprTuple	45, 28
26	Operation	operator: 48 operands: 29
27	Operation	operator: 30 operand: 45
28	Operation	operator: 32 operands: 33
29	ExprTuple	34, 35
30	Variable
31	ExprTuple	45
32	Literal
33	ExprTuple	36, 37
34	Literal
35	Operation	operator: 38 operands: 39
36	Literal
37	Operation	operator: 40 operands: 41
38	Literal
39	ExprTuple	52, 42, 43, 44, 45
40	Literal
41	ExprTuple	46, 47
42	Literal
43	Literal
44	Literal
45	Variable
46	Operation	operator: 48 operands: 49
47	Operation	operator: 50 operand: 54
48	Literal
49	ExprTuple	52, 53
50	Literal
51	ExprTuple	54
52	Literal
53	Literal
54	Literal