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 _m_domain, _phase, _t, _two_pow_t

# build up the expression from sub-expressions
sub_expr1 = frac(one, Exp(two, frac(_t, two)))
sub_expr2 = 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)
expr = Equals(Mult(sub_expr1, sub_expr1, sub_expr2), Mult(Exp(sub_expr1, two), sub_expr2)).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left(\frac{1}{2^{\frac{t}{2}}} \cdot \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 \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right)\right)\right) =  \\ \left(\left(\frac{1}{2^{\frac{t}{2}}}\right)^{2} \cdot \left(\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)\right) \end{array} \end{array}

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.	()	(2)	('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: 52 operands: 5
4	Operation	operator: 52 operands: 6
5	ExprTuple	12, 12, 8
6	ExprTuple	7, 8
7	Operation	operator: 54 operands: 9
8	Operation	operator: 10 operand: 13
9	ExprTuple	12, 60
10	Literal
11	ExprTuple	13
12	Operation	operator: 45 operands: 14
13	Lambda	parameter: 58 body: 16
14	ExprTuple	51, 17
15	ExprTuple	58
16	Conditional	value: 18 condition: 19
17	Operation	operator: 54 operands: 20
18	Operation	operator: 52 operands: 21
19	Operation	operator: 22 operands: 23
20	ExprTuple	60, 24
21	ExprTuple	25, 26
22	Literal
23	ExprTuple	58, 27
24	Operation	operator: 45 operands: 28
25	Operation	operator: 54 operands: 29
26	Operation	operator: 54 operands: 30
27	Operation	operator: 31 operands: 32
28	ExprTuple	61, 60
29	ExprTuple	34, 33
30	ExprTuple	34, 35
31	Literal
32	ExprTuple	36, 37
33	Operation	operator: 52 operands: 38
34	Literal
35	Operation	operator: 47 operand: 43
36	Literal
37	Operation	operator: 40 operands: 41
38	ExprTuple	60, 56, 57, 42, 58
39	ExprTuple	43
40	Literal
41	ExprTuple	50, 44
42	Literal
43	Operation	operator: 45 operands: 46
44	Operation	operator: 47 operand: 51
45	Literal
46	ExprTuple	49, 50
47	Literal
48	ExprTuple	51
49	Operation	operator: 52 operands: 53
50	Operation	operator: 54 operands: 55
51	Literal
52	Literal
53	ExprTuple	60, 56, 57, 58, 59
54	Literal
55	ExprTuple	60, 61
56	Literal
57	Literal
58	Variable
59	Variable
60	Literal
61	Literal