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 Function, Variable, 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 = Interval(zero, subtract(two_pow_t, one))
expr = Equals(VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr2, NumKet(k, Add(t, one))), domain = Interval(zero, subtract(Mult(two, two_pow_t), one))), VecAdd(VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr2, Function(Variable("_a", latex_format = r"{_{-}a}"), sub_expr1)), domain = sub_expr3), 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_expr3)))))

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 {_{-}a}\left(k\right)\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: 36 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	9, 10
8	Lambda	parameter: 74 body: 11
9	Operation	operator: 36 operand: 16
10	Operation	operator: 51 operands: 13
11	Conditional	value: 14 condition: 15
12	ExprTuple	16
13	ExprTuple	17, 18
14	Operation	operator: 51 operands: 19
15	Operation	operator: 53 operands: 20
16	Lambda	parameter: 74 body: 21
17	Operation	operator: 77 operands: 22
18	Operation	operator: 23 operands: 24
19	ExprTuple	55, 25
20	ExprTuple	74, 26
21	Conditional	value: 27 condition: 49
22	ExprTuple	63, 28
23	Literal
24	ExprTuple	29, 30
25	Operation	operator: 59 operands: 31
26	Operation	operator: 61 operands: 32
27	Operation	operator: 51 operands: 33
28	Operation	operator: 67 operands: 34
29	Operation	operator: 35 operand: 83
30	Operation	operator: 36 operand: 41
31	ExprTuple	74, 38
32	ExprTuple	65, 39
33	ExprTuple	55, 40
34	ExprTuple	81, 71, 72, 73, 75
35	Literal
36	Literal
37	ExprTuple	41
38	Operation	operator: 69 operands: 42
39	Operation	operator: 69 operands: 43
40	Operation	operator: 44 operand: 74
41	Lambda	parameter: 74 body: 46
42	ExprTuple	82, 83
43	ExprTuple	47, 76
44	Variable
45	ExprTuple	74
46	Conditional	value: 48 condition: 49
47	Operation	operator: 67 operands: 50
48	Operation	operator: 51 operands: 52
49	Operation	operator: 53 operands: 54
50	ExprTuple	81, 75
51	Literal
52	ExprTuple	55, 56
53	Literal
54	ExprTuple	74, 57
55	Operation	operator: 77 operands: 58
56	Operation	operator: 59 operands: 60
57	Operation	operator: 61 operands: 62
58	ExprTuple	63, 64
59	Literal
60	ExprTuple	74, 82
61	Literal
62	ExprTuple	65, 66
63	Literal
64	Operation	operator: 67 operands: 68
65	Literal
66	Operation	operator: 69 operands: 70
67	Literal
68	ExprTuple	81, 71, 72, 73, 74
69	Literal
70	ExprTuple	75, 76
71	Literal
72	Literal
73	Literal
74	Variable
75	Operation	operator: 77 operands: 78
76	Operation	operator: 79 operand: 83
77	Literal
78	ExprTuple	81, 82
79	Literal
80	ExprTuple	83
81	Literal
82	Variable
83	Literal