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 Variable, k, t
from proveit.linear_algebra import ScalarMult, VecAdd, VecSum
from proveit.logic import Equals
from proveit.numbers import Add, Exp, Interval, Mult, e, i, one, pi, subtract, two, zero
from proveit.physics.quantum import NumKet
from proveit.physics.quantum.QPE import _phase, two_pow_t

# build up the expression from sub-expressions
sub_expr1 = [k]
sub_expr2 = ScalarMult(Exp(e, Mult(two, pi, i, _phase, k)), NumKet(k, Add(t, one)))
expr = Equals(VecSum(index_or_indices = sub_expr1, summand = sub_expr2, domain = Interval(zero, subtract(Mult(two, two_pow_t), one))), VecAdd(VecSum(index_or_indices = sub_expr1, summand = sub_expr2, domain = Interval(zero, subtract(two_pow_t, one))), Variable("_a", latex_format = r"{_{-}a}")))

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