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_mod_two_pow_t, _m_domain, _phase, _two_pow_t

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