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 Conditional, ConditionalSet, ExprRange, Variable, t
from proveit.core_expr_types import Len
from proveit.linear_algebra import MatrixExp
from proveit.logic import Equals, NotEquals, Set
from proveit.numbers import Add, Exp, Interval, Neg, one, two, zero
from proveit.physics.quantum import CONTROL
from proveit.physics.quantum.QPE import _U, _s
from proveit.physics.quantum.circuits import Gate, Igate, MultiQubitElem

# build up the expression from sub-expressions
sub_expr1 = Variable("_b", latex_format = r"{_{-}b}")
sub_expr2 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr3 = Add(t, one)
sub_expr4 = Add(sub_expr1, t)
expr = Equals(Len(operands = [ExprRange(sub_expr1, [ExprRange(sub_expr2, ConditionalSet(Conditional(MultiQubitElem(element = CONTROL, targets = Set(sub_expr3)), Equals(sub_expr4, sub_expr2)), Conditional(Igate, NotEquals(sub_expr4, sub_expr2))), one, t).with_case_simplification(), ExprRange(sub_expr2, MultiQubitElem(element = Gate(operation = MatrixExp(_U, Exp(two, Neg(sub_expr1))), part = sub_expr2), targets = Interval(sub_expr3, Add(t, _s))), one, _s)], Add(Neg(t), one), zero).with_decreasing_order()]), Len(operands = [ExprRange(sub_expr2, sub_expr2, one, t)]))

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(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{CONTROL~\mbox{on}~\left\{t + 1\right\}} & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{U^{2^{t - 1}}~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{U^{2^{t - 1}}~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{U^{2^{t - 1}}~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}\right), \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{CONTROL~\mbox{on}~\left\{t + 1\right\}} & \qw 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{U^{2^{t - 2}}~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{U^{2^{t - 2}}~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{U^{2^{t - 2}}~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}\right), \ldots, \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \qw & \qw 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{CONTROL~\mbox{on}~\left\{t + 1\right\}} & \qw 
} \end{array},\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{U^{2^{0}}~\mbox{part}~1~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{U^{2^{0}}~\mbox{part}~2~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}, \ldots, \begin{array}{c} \Qcircuit@C=1em @R=.7em{
& & \gate{U^{2^{0}}~\mbox{part}~s~\mbox{on}~\{t + 1~\ldotp \ldotp~t + s\}} & \qw 
} \end{array}\right)\right)| = |\left(1, 2, \ldots, t\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: 40 operands: 1
1	ExprTuple	2, 3
2	Operation	operator: 5 operands: 4
3	Operation	operator: 5 operands: 6
4	ExprTuple	7
5	Literal
6	ExprTuple	8
7	ExprRange	lambda_map: 9 start_index: 10 end_index: 11
8	ExprRange	lambda_map: 12 start_index: 69 end_index: 68
9	Lambda	parameter: 72 body: 13
10	Operation	operator: 64 operands: 14
11	Literal
12	Lambda	parameter: 51 body: 51
13	ExprTuple	15, 16
14	ExprTuple	17, 69
15	ExprRange	lambda_map: 18 start_index: 69 end_index: 68
16	ExprRange	lambda_map: 19 start_index: 69 end_index: 60
17	Operation	operator: 70 operand: 68
18	Lambda	parameter: 51 body: 21
19	Lambda	parameter: 51 body: 23
20	ExprTuple	68
21	Operation	operator: 24 operands: 25
22	ExprTuple	51
23	Operation	operator: 38 operands: 26
24	Literal
25	ExprTuple	27, 28
26	NamedExprs	element: 29 targets: 30
27	Conditional	value: 31 condition: 32
28	Conditional	value: 33 condition: 34
29	Operation	operator: 41 operands: 35
30	Operation	operator: 36 operands: 37
31	Operation	operator: 38 operands: 39
32	Operation	operator: 40 operands: 44
33	Operation	operator: 41 operands: 42
34	Operation	operator: 43 operands: 44
35	NamedExprs	operation: 45 part: 51
36	Literal
37	ExprTuple	61, 46
38	Literal
39	NamedExprs	element: 47 targets: 48
40	Literal
41	Literal
42	NamedExprs	operation: 49
43	Literal
44	ExprTuple	50, 51
45	Operation	operator: 52 operands: 53
46	Operation	operator: 64 operands: 54
47	Literal
48	Operation	operator: 55 operand: 61
49	Literal
50	Operation	operator: 64 operands: 57
51	Variable
52	Literal
53	ExprTuple	58, 59
54	ExprTuple	68, 60
55	Literal
56	ExprTuple	61
57	ExprTuple	72, 68
58	Literal
59	Operation	operator: 62 operands: 63
60	Literal
61	Operation	operator: 64 operands: 65
62	Literal
63	ExprTuple	66, 67
64	Literal
65	ExprTuple	68, 69
66	Literal
67	Operation	operator: 70 operand: 72
68	Variable
69	Literal
70	Literal
71	ExprTuple	72
72	Variable