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, 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)))
sub_expr3 = subtract(Mult(two, two_pow_t), one)
expr = Equals(VecSum(index_or_indices = sub_expr1, summand = sub_expr2, domain = Interval(zero, sub_expr3)), VecAdd(VecSum(index_or_indices = sub_expr1, summand = sub_expr2, domain = Interval(zero, subtract(two_pow_t, one))), VecSum(index_or_indices = sub_expr1, summand = sub_expr2, domain = Interval(two_pow_t, 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 \lvert k \rangle_{t + 1}\right)\right) + \left(\sum_{k=2^{t}}^{\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)\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: 13 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	9, 10
8	Lambda	parameter: 56 body: 11
9	Operation	operator: 13 operand: 16
10	Operation	operator: 13 operand: 17
11	Conditional	value: 24 condition: 15
12	ExprTuple	16
13	Literal
14	ExprTuple	17
15	Operation	operator: 30 operands: 18
16	Lambda	parameter: 56 body: 19
17	Lambda	parameter: 56 body: 21
18	ExprTuple	56, 22
19	Conditional	value: 24 condition: 23
20	ExprTuple	56
21	Conditional	value: 24 condition: 25
22	Operation	operator: 40 operands: 26
23	Operation	operator: 30 operands: 27
24	Operation	operator: 28 operands: 29
25	Operation	operator: 30 operands: 31
26	ExprTuple	42, 47
27	ExprTuple	56, 32
28	Literal
29	ExprTuple	33, 34
30	Literal
31	ExprTuple	56, 35
32	Operation	operator: 40 operands: 36
33	Operation	operator: 65 operands: 37
34	Operation	operator: 38 operands: 39
35	Operation	operator: 40 operands: 41
36	ExprTuple	42, 43
37	ExprTuple	44, 45
38	Literal
39	ExprTuple	56, 46
40	Literal
41	ExprTuple	63, 47
42	Literal
43	Operation	operator: 51 operands: 48
44	Literal
45	Operation	operator: 59 operands: 49
46	Operation	operator: 51 operands: 50
47	Operation	operator: 51 operands: 52
48	ExprTuple	63, 58
49	ExprTuple	67, 53, 54, 55, 56
50	ExprTuple	68, 64
51	Literal
52	ExprTuple	57, 58
53	Literal
54	Literal
55	Literal
56	Variable
57	Operation	operator: 59 operands: 60
58	Operation	operator: 61 operand: 64
59	Literal
60	ExprTuple	67, 63
61	Literal
62	ExprTuple	64
63	Operation	operator: 65 operands: 66
64	Literal
65	Literal
66	ExprTuple	67, 68
67	Literal
68	Variable