Expression of type Conditional

from the theory of proveit.physics.quantum.circuits

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
from proveit.core_expr_types.expr_arrays import A11_to_Akl
from proveit.logic import Equals, Forall, InSet
from proveit.physics.quantum import m_ket_domain, var_ket_psi
from proveit.physics.quantum.circuits import circuit__psi_m__u_Akl_v, circuit__u_Akl_v
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [A11_to_Akl], instance_expr = Equals(Prob(circuit__u_Akl_v), Prob(circuit__psi_m__u_Akl_v)).with_wrapping_at(2)).with_wrapping(), InSet(var_ket_psi, m_ket_domain))

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\{\begin{array}{l}\forall_{A_{1, 1}, A_{1, 2}, \ldots, A_{1, l}, A_{2, 1}, A_{2, 2}, \ldots, A_{2, l}, \ldots\ldots, A_{k, 1}, A_{k, 2}, \ldots, A_{k, l}}~\\
\left(\begin{array}{c} \begin{array}{l} \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\multiqin{3}{\lvert u \rangle} & \gate{A_{1, 1}} \qwx[1] & \gate{A_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 1}} \qwx[1] & \multiqout{3}{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, 2}} \qwx[1] & \gate{A_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 2}} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, l}} & \gate{A_{2, l}} & \gate{\cdots} & \gate{A_{k, l}} & \ghostqout{\lvert v \rangle}
} \end{array}\right) =  \\ \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert \psi \rangle} & { /^{m} } \qw & { /^{m} } \qw & \gate{\cdots} & { /^{m} } \qw & \qout{\lvert \psi \rangle} \\
\multiqin{3}{\lvert u \rangle} & \gate{A_{1, 1}} \qwx[1] & \gate{A_{2, 1}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 1}} \qwx[1] & \multiqout{3}{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, 2}} \qwx[1] & \gate{A_{2, 2}} \qwx[1] & \gate{\cdots} \qwx[1] & \gate{A_{k, 2}} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{\vdots} \qwx[1] & \gate{\vdots} \qwx[1] & \gate{\ddots} \qwx[1] & \gate{\vdots} \qwx[1] & \ghostqout{\lvert v \rangle} \\
\ghostqin{\lvert u \rangle} & \gate{A_{1, l}} & \gate{A_{2, l}} & \gate{\cdots} & \gate{A_{k, l}} & \ghostqout{\lvert v \rangle}
} \end{array}\right) \end{array} \end{array}\right)\end{array} \textrm{ if } \lvert \psi \rangle \in \mathbb{C}^{2^{m}}\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 7
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7
5	Literal
6	ExprTuple	99, 8
7	Lambda	parameters: 9 body: 10
8	Operation	operator: 11 operands: 12
9	ExprTuple	13
10	Operation	operator: 14 operands: 15
11	Literal
12	ExprTuple	16, 17
13	ExprRange	lambda_map: 18 start_index: 107 end_index: 44
14	Literal
15	ExprTuple	19, 20
16	Literal
17	Operation	operator: 21 operands: 22
18	Lambda	parameter: 98 body: 61
19	Operation	operator: 24 operand: 27
20	Operation	operator: 24 operand: 28
21	Literal
22	ExprTuple	26, 108
23	ExprTuple	27
24	Literal
25	ExprTuple	28
26	Literal
27	Operation	operator: 30 operands: 29
28	Operation	operator: 30 operands: 31
29	ExprTuple	32, 33, 34
30	Literal
31	ExprTuple	35, 36, 37
32	ExprTuple	38
33	ExprRange	lambda_map: 39 start_index: 107 end_index: 44
34	ExprTuple	40
35	ExprTuple	41, 42
36	ExprRange	lambda_map: 43 start_index: 107 end_index: 44
37	ExprTuple	45, 46
38	ExprRange	lambda_map: 47 start_index: 107 end_index: 109
39	Lambda	parameter: 98 body: 48
40	ExprRange	lambda_map: 49 start_index: 107 end_index: 109
41	ExprRange	lambda_map: 50 start_index: 107 end_index: 108
42	ExprRange	lambda_map: 51 start_index: 107 end_index: 109
43	Lambda	parameter: 98 body: 53
44	Variable
45	ExprRange	lambda_map: 54 start_index: 107 end_index: 108
46	ExprRange	lambda_map: 55 start_index: 107 end_index: 109
47	Lambda	parameter: 101 body: 56
48	ExprTuple	61
49	Lambda	parameter: 101 body: 57
50	Lambda	parameter: 101 body: 58
51	Lambda	parameter: 101 body: 59
52	ExprTuple	98
53	ExprTuple	60, 61
54	Lambda	parameter: 101 body: 62
55	Lambda	parameter: 101 body: 63
56	Operation	operator: 71 operands: 64
57	Operation	operator: 71 operands: 65
58	Operation	operator: 71 operands: 66
59	Operation	operator: 71 operands: 67
60	ExprRange	lambda_map: 68 start_index: 107 end_index: 108
61	ExprRange	lambda_map: 69 start_index: 107 end_index: 109
62	Operation	operator: 71 operands: 70
63	Operation	operator: 71 operands: 72
64	NamedExprs	element: 75 targets: 73
65	NamedExprs	element: 81 targets: 73
66	NamedExprs	element: 74 targets: 80
67	NamedExprs	element: 75 targets: 82
68	Lambda	parameter: 101 body: 76
69	Lambda	parameter: 101 body: 78
70	NamedExprs	element: 79 targets: 80
71	Literal
72	NamedExprs	element: 81 targets: 82
73	Operation	operator: 94 operands: 83
74	Operation	operator: 84 operands: 90
75	Operation	operator: 84 operands: 85
76	Operation	operator: 86 operands: 87
77	ExprTuple	101
78	IndexedVar	variable: 88 indices: 89
79	Operation	operator: 92 operands: 90
80	Operation	operator: 94 operands: 91
81	Operation	operator: 92 operands: 93
82	Operation	operator: 94 operands: 95
83	ExprTuple	107, 109
84	Literal
85	NamedExprs	state: 96 part: 101
86	Literal
87	NamedExprs	operation: 97
88	Variable
89	ExprTuple	98, 101
90	NamedExprs	state: 99 part: 101
91	ExprTuple	107, 108
92	Literal
93	NamedExprs	state: 100 part: 101
94	Literal
95	ExprTuple	102, 103
96	Variable
97	Literal
98	Variable
99	Variable
100	Variable
101	Variable
102	Operation	operator: 105 operands: 104
103	Operation	operator: 105 operands: 106
104	ExprTuple	108, 107
105	Literal
106	ExprTuple	108, 109
107	Literal
108	Variable
109	Variable