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
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, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = frac(one, _two_pow_t)
expr = Equals(Mult(sub_expr1, Sum(index_or_indices = [k], summand = Mult(Exp(e, Neg(frac(Mult(two, pi, i, k, Mult(_two_pow_t, _phase)), _two_pow_t))), Exp(e, Mult(two, pi, i, _phase, k))), domain = _m_domain)), Mult(sub_expr1, _two_pow_t))

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