Expression of type ExprTuple

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, ExprTuple, Lambda, k, l, m
from proveit.core_expr_types.expr_arrays import A11_to_Akl
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import NaturalPos
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__psi_m__u_Akl_v, circuit__u_Akl_v
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = ExprTuple(Lambda([k, l, m], Conditional(Forall(instance_param_or_params = [var_ket_u, var_ket_v], instance_expr = 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__psi_m__u_Akl_v)).with_wrapping_at(2)).with_wrapping(), domain = m_ket_domain), domain = l_ket_domain), And(InSet(k, NaturalPos), InSet(l, NaturalPos), InSet(m, NaturalPos)))))

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(k, l, m\right) \mapsto \left\{\forall_{\lvert u \rangle, \lvert v \rangle \in \mathbb{C}^{2^{l}}}~\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{
\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}\right]\right] \textrm{ if } k \in \mathbb{N}^+ ,  l \in \mathbb{N}^+ ,  m \in \mathbb{N}^+\right..\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	ExprTuple	1
1	Lambda	parameters: 2 body: 3
2	ExprTuple	77, 142, 141
3	Conditional	value: 4 condition: 5
4	Operation	operator: 33 operand: 8
5	Operation	operator: 21 operands: 7
6	ExprTuple	8
7	ExprTuple	9, 10, 11
8	Lambda	parameters: 12 body: 13
9	Operation	operator: 35 operands: 14
10	Operation	operator: 35 operands: 15
11	Operation	operator: 35 operands: 16
12	ExprTuple	129, 133
13	Conditional	value: 17 condition: 18
14	ExprTuple	77, 19
15	ExprTuple	142, 19
16	ExprTuple	141, 19
17	Operation	operator: 33 operand: 23
18	Operation	operator: 21 operands: 22
19	Literal
20	ExprTuple	23
21	Literal
22	ExprTuple	24, 25
23	Lambda	parameter: 132 body: 27
24	Operation	operator: 35 operands: 28
25	Operation	operator: 35 operands: 29
26	ExprTuple	132
27	Conditional	value: 30 condition: 31
28	ExprTuple	129, 32
29	ExprTuple	133, 32
30	Operation	operator: 33 operand: 38
31	Operation	operator: 35 operands: 36
32	Operation	operator: 43 operands: 37
33	Literal
34	ExprTuple	38
35	Literal
36	ExprTuple	132, 39
37	ExprTuple	49, 40
38	Lambda	parameters: 41 body: 42
39	Operation	operator: 43 operands: 44
40	Operation	operator: 54 operands: 45
41	ExprTuple	46
42	Operation	operator: 47 operands: 48
43	Literal
44	ExprTuple	49, 50
45	ExprTuple	59, 142
46	ExprRange	lambda_map: 51 start_index: 140 end_index: 77
47	Literal
48	ExprTuple	52, 53
49	Literal
50	Operation	operator: 54 operands: 55
51	Lambda	parameter: 131 body: 94
52	Operation	operator: 57 operand: 60
53	Operation	operator: 57 operand: 61
54	Literal
55	ExprTuple	59, 141
56	ExprTuple	60
57	Literal
58	ExprTuple	61
59	Literal
60	Operation	operator: 63 operands: 62
61	Operation	operator: 63 operands: 64
62	ExprTuple	65, 66, 67
63	Literal
64	ExprTuple	68, 69, 70
65	ExprTuple	71
66	ExprRange	lambda_map: 72 start_index: 140 end_index: 77
67	ExprTuple	73
68	ExprTuple	74, 75
69	ExprRange	lambda_map: 76 start_index: 140 end_index: 77
70	ExprTuple	78, 79
71	ExprRange	lambda_map: 80 start_index: 140 end_index: 142
72	Lambda	parameter: 131 body: 81
73	ExprRange	lambda_map: 82 start_index: 140 end_index: 142
74	ExprRange	lambda_map: 83 start_index: 140 end_index: 141
75	ExprRange	lambda_map: 84 start_index: 140 end_index: 142
76	Lambda	parameter: 131 body: 86
77	Variable
78	ExprRange	lambda_map: 87 start_index: 140 end_index: 141
79	ExprRange	lambda_map: 88 start_index: 140 end_index: 142
80	Lambda	parameter: 134 body: 89
81	ExprTuple	94
82	Lambda	parameter: 134 body: 90
83	Lambda	parameter: 134 body: 91
84	Lambda	parameter: 134 body: 92
85	ExprTuple	131
86	ExprTuple	93, 94
87	Lambda	parameter: 134 body: 95
88	Lambda	parameter: 134 body: 96
89	Operation	operator: 104 operands: 97
90	Operation	operator: 104 operands: 98
91	Operation	operator: 104 operands: 99
92	Operation	operator: 104 operands: 100
93	ExprRange	lambda_map: 101 start_index: 140 end_index: 141
94	ExprRange	lambda_map: 102 start_index: 140 end_index: 142
95	Operation	operator: 104 operands: 103
96	Operation	operator: 104 operands: 105
97	NamedExprs	element: 108 targets: 106
98	NamedExprs	element: 114 targets: 106
99	NamedExprs	element: 107 targets: 113
100	NamedExprs	element: 108 targets: 115
101	Lambda	parameter: 134 body: 109
102	Lambda	parameter: 134 body: 111
103	NamedExprs	element: 112 targets: 113
104	Literal
105	NamedExprs	element: 114 targets: 115
106	Operation	operator: 127 operands: 116
107	Operation	operator: 117 operands: 123
108	Operation	operator: 117 operands: 118
109	Operation	operator: 119 operands: 120
110	ExprTuple	134
111	IndexedVar	variable: 121 indices: 122
112	Operation	operator: 125 operands: 123
113	Operation	operator: 127 operands: 124
114	Operation	operator: 125 operands: 126
115	Operation	operator: 127 operands: 128
116	ExprTuple	140, 142
117	Literal
118	NamedExprs	state: 129 part: 134
119	Literal
120	NamedExprs	operation: 130
121	Variable
122	ExprTuple	131, 134
123	NamedExprs	state: 132 part: 134
124	ExprTuple	140, 141
125	Literal
126	NamedExprs	state: 133 part: 134
127	Literal
128	ExprTuple	135, 136
129	Variable
130	Literal
131	Variable
132	Variable
133	Variable
134	Variable
135	Operation	operator: 138 operands: 137
136	Operation	operator: 138 operands: 139
137	ExprTuple	141, 140
138	Literal
139	ExprTuple	141, 142
140	Literal
141	Variable
142	Variable