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