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 = [k]
sub_expr2 = Exp(e, Mult(two, pi, i, _phase, k))
sub_expr3 = frac(one, Exp(two, frac(_t, two)))
sub_expr4 = Exp(e, Neg(frac(Mult(two, pi, i, k, m), _two_pow_t)))
expr = Equals(Mult(sub_expr3, Sum(index_or_indices = sub_expr1, summand = Mult(sub_expr2, Mult(sub_expr3, sub_expr4)), domain = _m_domain)), Mult(Exp(sub_expr3, two), Sum(index_or_indices = sub_expr1, summand = Mult(sub_expr2, sub_expr4), 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())

\left(\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 \left(\frac{1}{2^{\frac{t}{2}}} \cdot \mathsf{e}^{-\frac{2 \cdot \pi \cdot \mathsf{i} \cdot k \cdot m}{2^{t}}}\right)\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)

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: 60 operands: 5
4	Operation	operator: 60 operands: 6
5	ExprTuple	33, 7
6	ExprTuple	8, 9
7	Operation	operator: 12 operand: 14
8	Operation	operator: 62 operands: 11
9	Operation	operator: 12 operand: 15
10	ExprTuple	14
11	ExprTuple	33, 68
12	Literal
13	ExprTuple	15
14	Lambda	parameter: 66 body: 16
15	Lambda	parameter: 66 body: 18
16	Conditional	value: 19 condition: 21
17	ExprTuple	66
18	Conditional	value: 20 condition: 21
19	Operation	operator: 60 operands: 22
20	Operation	operator: 60 operands: 23
21	Operation	operator: 24 operands: 25
22	ExprTuple	27, 26
23	ExprTuple	27, 34
24	Literal
25	ExprTuple	66, 28
26	Operation	operator: 60 operands: 29
27	Operation	operator: 62 operands: 30
28	Operation	operator: 31 operands: 32
29	ExprTuple	33, 34
30	ExprTuple	44, 35
31	Literal
32	ExprTuple	36, 37
33	Operation	operator: 56 operands: 38
34	Operation	operator: 62 operands: 39
35	Operation	operator: 60 operands: 40
36	Literal
37	Operation	operator: 41 operands: 42
38	ExprTuple	54, 43
39	ExprTuple	44, 45
40	ExprTuple	68, 64, 65, 46, 66
41	Literal
42	ExprTuple	59, 47
43	Operation	operator: 62 operands: 48
44	Literal
45	Operation	operator: 50 operand: 53
46	Literal
47	Operation	operator: 50 operand: 54
48	ExprTuple	68, 52
49	ExprTuple	53
50	Literal
51	ExprTuple	54
52	Operation	operator: 56 operands: 55
53	Operation	operator: 56 operands: 57
54	Literal
55	ExprTuple	69, 68
56	Literal
57	ExprTuple	58, 59
58	Operation	operator: 60 operands: 61
59	Operation	operator: 62 operands: 63
60	Literal
61	ExprTuple	68, 64, 65, 66, 67
62	Literal
63	ExprTuple	68, 69
64	Literal
65	Literal
66	Variable
67	Variable
68	Literal
69	Literal