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__u_Akl_v, circuit__u_Akl_v__psi_m
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__u_Akl_v__psi_m)).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{
\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} \\
\qin{\lvert \psi \rangle} & { /^{m} } \qw & { /^{m} } \qw & \gate{\cdots} & { /^{m} } \qw & \qout{\lvert \psi \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	90, 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: 97 end_index: 42
14	Literal
15	ExprTuple	19, 20
16	Literal
17	Operation	operator: 21 operands: 22
18	Lambda	parameter: 87 body: 54
19	Operation	operator: 24 operand: 27
20	Operation	operator: 24 operand: 28
21	Literal
22	ExprTuple	26, 99
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	39
33	ExprRange	lambda_map: 38 start_index: 97 end_index: 42
34	ExprTuple	43
35	ExprTuple	39, 40
36	ExprRange	lambda_map: 41 start_index: 97 end_index: 42
37	ExprTuple	43, 44
38	Lambda	parameter: 87 body: 45
39	ExprRange	lambda_map: 46 start_index: 97 end_index: 98
40	ExprRange	lambda_map: 47 start_index: 97 end_index: 99
41	Lambda	parameter: 87 body: 49
42	Variable
43	ExprRange	lambda_map: 50 start_index: 97 end_index: 98
44	ExprRange	lambda_map: 51 start_index: 97 end_index: 99
45	ExprTuple	54
46	Lambda	parameter: 91 body: 52
47	Lambda	parameter: 91 body: 53
48	ExprTuple	87
49	ExprTuple	54, 55
50	Lambda	parameter: 91 body: 56
51	Lambda	parameter: 91 body: 57
52	Operation	operator: 63 operands: 58
53	Operation	operator: 63 operands: 59
54	ExprRange	lambda_map: 60 start_index: 97 end_index: 98
55	ExprRange	lambda_map: 61 start_index: 97 end_index: 99
56	Operation	operator: 63 operands: 62
57	Operation	operator: 63 operands: 64
58	NamedExprs	element: 65 targets: 71
59	NamedExprs	element: 66 targets: 73
60	Lambda	parameter: 91 body: 67
61	Lambda	parameter: 91 body: 69
62	NamedExprs	element: 70 targets: 71
63	Literal
64	NamedExprs	element: 72 targets: 73
65	Operation	operator: 75 operands: 74
66	Operation	operator: 75 operands: 83
67	IndexedVar	variable: 76 indices: 77
68	ExprTuple	91
69	Operation	operator: 78 operands: 79
70	Operation	operator: 82 operands: 80
71	Operation	operator: 84 operands: 81
72	Operation	operator: 82 operands: 83
73	Operation	operator: 84 operands: 85
74	NamedExprs	state: 86 part: 91
75	Literal
76	Variable
77	ExprTuple	87, 91
78	Literal
79	NamedExprs	operation: 88
80	NamedExprs	state: 89 part: 91
81	ExprTuple	97, 98
82	Literal
83	NamedExprs	state: 90 part: 91
84	Literal
85	ExprTuple	92, 93
86	Variable
87	Variable
88	Literal
89	Variable
90	Variable
91	Variable
92	Operation	operator: 95 operands: 94
93	Operation	operator: 95 operands: 96
94	ExprTuple	98, 97
95	Literal
96	ExprTuple	98, 99
97	Literal
98	Variable
99	Variable