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, Mult(sub_expr1, sub_expr2)), Mult(Exp(sub_expr1, two), sub_expr2))

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