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 And, Equals, Forall, InSet
from proveit.physics.quantum import l_ket_domain, m_ket_domain, var_ket_psi, var_ket_u, var_ket_v
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 = [var_ket_psi], instance_expr = 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(), domain = m_ket_domain), And(InSet(var_ket_u, l_ket_domain), InSet(var_ket_v, l_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\{\forall_{\lvert \psi \rangle \in \mathbb{C}^{2^{m}}}~\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}\right] \textrm{ if } \lvert u \rangle \in \mathbb{C}^{2^{l}} ,  \lvert v \rangle \in \mathbb{C}^{2^{l}}\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: 16 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	7, 8
6	Lambda	parameter: 106 body: 10
7	Operation	operator: 18 operands: 11
8	Operation	operator: 18 operands: 12
9	ExprTuple	106
10	Conditional	value: 13 condition: 14
11	ExprTuple	102, 15
12	ExprTuple	105, 15
13	Operation	operator: 16 operand: 21
14	Operation	operator: 18 operands: 19
15	Operation	operator: 26 operands: 20
16	Literal
17	ExprTuple	21
18	Literal
19	ExprTuple	106, 22
20	ExprTuple	32, 23
21	Lambda	parameters: 24 body: 25
22	Operation	operator: 26 operands: 27
23	Operation	operator: 37 operands: 28
24	ExprTuple	29
25	Operation	operator: 30 operands: 31
26	Literal
27	ExprTuple	32, 33
28	ExprTuple	42, 114
29	ExprRange	lambda_map: 34 start_index: 113 end_index: 58
30	Literal
31	ExprTuple	35, 36
32	Literal
33	Operation	operator: 37 operands: 38
34	Lambda	parameter: 103 body: 70
35	Operation	operator: 40 operand: 43
36	Operation	operator: 40 operand: 44
37	Literal
38	ExprTuple	42, 115
39	ExprTuple	43
40	Literal
41	ExprTuple	44
42	Literal
43	Operation	operator: 46 operands: 45
44	Operation	operator: 46 operands: 47
45	ExprTuple	48, 49, 50
46	Literal
47	ExprTuple	51, 52, 53
48	ExprTuple	55
49	ExprRange	lambda_map: 54 start_index: 113 end_index: 58
50	ExprTuple	59
51	ExprTuple	55, 56
52	ExprRange	lambda_map: 57 start_index: 113 end_index: 58
53	ExprTuple	59, 60
54	Lambda	parameter: 103 body: 61
55	ExprRange	lambda_map: 62 start_index: 113 end_index: 114
56	ExprRange	lambda_map: 63 start_index: 113 end_index: 115
57	Lambda	parameter: 103 body: 65
58	Variable
59	ExprRange	lambda_map: 66 start_index: 113 end_index: 114
60	ExprRange	lambda_map: 67 start_index: 113 end_index: 115
61	ExprTuple	70
62	Lambda	parameter: 107 body: 68
63	Lambda	parameter: 107 body: 69
64	ExprTuple	103
65	ExprTuple	70, 71
66	Lambda	parameter: 107 body: 72
67	Lambda	parameter: 107 body: 73
68	Operation	operator: 79 operands: 74
69	Operation	operator: 79 operands: 75
70	ExprRange	lambda_map: 76 start_index: 113 end_index: 114
71	ExprRange	lambda_map: 77 start_index: 113 end_index: 115
72	Operation	operator: 79 operands: 78
73	Operation	operator: 79 operands: 80
74	NamedExprs	element: 81 targets: 87
75	NamedExprs	element: 82 targets: 89
76	Lambda	parameter: 107 body: 83
77	Lambda	parameter: 107 body: 85
78	NamedExprs	element: 86 targets: 87
79	Literal
80	NamedExprs	element: 88 targets: 89
81	Operation	operator: 91 operands: 90
82	Operation	operator: 91 operands: 99
83	IndexedVar	variable: 92 indices: 93
84	ExprTuple	107
85	Operation	operator: 94 operands: 95
86	Operation	operator: 98 operands: 96
87	Operation	operator: 100 operands: 97
88	Operation	operator: 98 operands: 99
89	Operation	operator: 100 operands: 101
90	NamedExprs	state: 102 part: 107
91	Literal
92	Variable
93	ExprTuple	103, 107
94	Literal
95	NamedExprs	operation: 104
96	NamedExprs	state: 105 part: 107
97	ExprTuple	113, 114
98	Literal
99	NamedExprs	state: 106 part: 107
100	Literal
101	ExprTuple	108, 109
102	Variable
103	Variable
104	Literal
105	Variable
106	Variable
107	Variable
108	Operation	operator: 111 operands: 110
109	Operation	operator: 111 operands: 112
110	ExprTuple	114, 113
111	Literal
112	ExprTuple	114, 115
113	Literal
114	Variable
115	Variable