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 ExprRange, Variable, k, m
from proveit.core_expr_types import Len
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 _m_domain, _phase, _t, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Equals(Len(operands = [frac(one, Exp(two, frac(_t, two))), Sum(index_or_indices = [k], summand = Mult(Exp(e, Mult(two, pi, i, _phase, k)), Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t)))), domain = _m_domain)]), Len(operands = [ExprRange(sub_expr1, sub_expr1, one, two)]))

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())

|\left(\frac{1}{2^{\frac{t}{2}}}, \sum_{k = 0}^{2^{t} - 1} \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot k} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right)\right)| = |\left(1, \ldots, 2\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: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 9
6	Literal
7	ExprTuple	10
8	Operation	operator: 48 operands: 11
9	Operation	operator: 12 operand: 16
10	ExprRange	lambda_map: 14 start_index: 54 end_index: 63
11	ExprTuple	54, 15
12	Literal
13	ExprTuple	16
14	Lambda	parameter: 21 body: 21
15	Operation	operator: 57 operands: 18
16	Lambda	parameter: 61 body: 20
17	ExprTuple	21
18	ExprTuple	63, 22
19	ExprTuple	61
20	Conditional	value: 23 condition: 24
21	Variable
22	Operation	operator: 48 operands: 25
23	Operation	operator: 55 operands: 26
24	Operation	operator: 27 operands: 28
25	ExprTuple	64, 63
26	ExprTuple	29, 30
27	Literal
28	ExprTuple	61, 31
29	Operation	operator: 57 operands: 32
30	Operation	operator: 57 operands: 33
31	Operation	operator: 34 operands: 35
32	ExprTuple	37, 36
33	ExprTuple	37, 38
34	Literal
35	ExprTuple	39, 40
36	Operation	operator: 55 operands: 41
37	Literal
38	Operation	operator: 50 operand: 46
39	Literal
40	Operation	operator: 43 operands: 44
41	ExprTuple	63, 59, 60, 45, 61
42	ExprTuple	46
43	Literal
44	ExprTuple	53, 47
45	Literal
46	Operation	operator: 48 operands: 49
47	Operation	operator: 50 operand: 54
48	Literal
49	ExprTuple	52, 53
50	Literal
51	ExprTuple	54
52	Operation	operator: 55 operands: 56
53	Operation	operator: 57 operands: 58
54	Literal
55	Literal
56	ExprTuple	63, 59, 60, 61, 62
57	Literal
58	ExprTuple	63, 64
59	Literal
60	Literal
61	Variable
62	Variable
63	Literal
64	Literal