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, t
from proveit.linear_algebra import ScalarMult, TensorProd, 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, ket1
from proveit.physics.quantum.QPE import _phase, 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 = ScalarMult(sub_expr2, NumKet(k, Add(t, one)))
sub_expr4 = Interval(zero, subtract(two_pow_t, one))
expr = Equals(VecSum(index_or_indices = sub_expr1, summand = sub_expr3, domain = Interval(zero, subtract(Mult(two, two_pow_t), one))), VecAdd(VecSum(index_or_indices = sub_expr1, summand = sub_expr3, domain = sub_expr4), ScalarMult(Exp(e, Mult(two, pi, i, _phase, two_pow_t)), TensorProd(ket1, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr2, NumKet(k, t)), domain = sub_expr4)))))

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) + \left(\mathsf{e}^{2 \cdot \pi \cdot \mathsf{i} \cdot \varphi \cdot 2^{t}} \cdot \left(\lvert 1 \rangle {\otimes} \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}\right)\right)\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: 32 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	9, 10
8	Lambda	parameter: 70 body: 11
9	Operation	operator: 32 operand: 15
10	Operation	operator: 47 operands: 13
11	Conditional	value: 24 condition: 14
12	ExprTuple	15
13	ExprTuple	16, 17
14	Operation	operator: 49 operands: 18
15	Lambda	parameter: 70 body: 19
16	Operation	operator: 73 operands: 20
17	Operation	operator: 21 operands: 22
18	ExprTuple	70, 23
19	Conditional	value: 24 condition: 44
20	ExprTuple	59, 25
21	Literal
22	ExprTuple	26, 27
23	Operation	operator: 57 operands: 28
24	Operation	operator: 47 operands: 29
25	Operation	operator: 63 operands: 30
26	Operation	operator: 31 operand: 79
27	Operation	operator: 32 operand: 36
28	ExprTuple	61, 34
29	ExprTuple	51, 35
30	ExprTuple	77, 67, 68, 69, 71
31	Literal
32	Literal
33	ExprTuple	36
34	Operation	operator: 65 operands: 37
35	Operation	operator: 55 operands: 38
36	Lambda	parameter: 70 body: 40
37	ExprTuple	41, 72
38	ExprTuple	70, 42
39	ExprTuple	70
40	Conditional	value: 43 condition: 44
41	Operation	operator: 63 operands: 45
42	Operation	operator: 65 operands: 46
43	Operation	operator: 47 operands: 48
44	Operation	operator: 49 operands: 50
45	ExprTuple	77, 71
46	ExprTuple	78, 79
47	Literal
48	ExprTuple	51, 52
49	Literal
50	ExprTuple	70, 53
51	Operation	operator: 73 operands: 54
52	Operation	operator: 55 operands: 56
53	Operation	operator: 57 operands: 58
54	ExprTuple	59, 60
55	Literal
56	ExprTuple	70, 78
57	Literal
58	ExprTuple	61, 62
59	Literal
60	Operation	operator: 63 operands: 64
61	Literal
62	Operation	operator: 65 operands: 66
63	Literal
64	ExprTuple	77, 67, 68, 69, 70
65	Literal
66	ExprTuple	71, 72
67	Literal
68	Literal
69	Literal
70	Variable
71	Operation	operator: 73 operands: 74
72	Operation	operator: 75 operand: 79
73	Literal
74	ExprTuple	77, 78
75	Literal
76	ExprTuple	79
77	Literal
78	Variable
79	Literal