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
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 = ExprTuple(Lambda([var_ket_u, var_ket_v], 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(\left(\lvert u \rangle, \lvert v \rangle\right) \mapsto \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..\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	105, 108
3	Conditional	value: 4 condition: 5
4	Operation	operator: 19 operand: 9
5	Operation	operator: 7 operands: 8
6	ExprTuple	9
7	Literal
8	ExprTuple	10, 11
9	Lambda	parameter: 109 body: 13
10	Operation	operator: 21 operands: 14
11	Operation	operator: 21 operands: 15
12	ExprTuple	109
13	Conditional	value: 16 condition: 17
14	ExprTuple	105, 18
15	ExprTuple	108, 18
16	Operation	operator: 19 operand: 24
17	Operation	operator: 21 operands: 22
18	Operation	operator: 29 operands: 23
19	Literal
20	ExprTuple	24
21	Literal
22	ExprTuple	109, 25
23	ExprTuple	35, 26
24	Lambda	parameters: 27 body: 28
25	Operation	operator: 29 operands: 30
26	Operation	operator: 40 operands: 31
27	ExprTuple	32
28	Operation	operator: 33 operands: 34
29	Literal
30	ExprTuple	35, 36
31	ExprTuple	45, 117
32	ExprRange	lambda_map: 37 start_index: 116 end_index: 61
33	Literal
34	ExprTuple	38, 39
35	Literal
36	Operation	operator: 40 operands: 41
37	Lambda	parameter: 106 body: 73
38	Operation	operator: 43 operand: 46
39	Operation	operator: 43 operand: 47
40	Literal
41	ExprTuple	45, 118
42	ExprTuple	46
43	Literal
44	ExprTuple	47
45	Literal
46	Operation	operator: 49 operands: 48
47	Operation	operator: 49 operands: 50
48	ExprTuple	51, 52, 53
49	Literal
50	ExprTuple	54, 55, 56
51	ExprTuple	58
52	ExprRange	lambda_map: 57 start_index: 116 end_index: 61
53	ExprTuple	62
54	ExprTuple	58, 59
55	ExprRange	lambda_map: 60 start_index: 116 end_index: 61
56	ExprTuple	62, 63
57	Lambda	parameter: 106 body: 64
58	ExprRange	lambda_map: 65 start_index: 116 end_index: 117
59	ExprRange	lambda_map: 66 start_index: 116 end_index: 118
60	Lambda	parameter: 106 body: 68
61	Variable
62	ExprRange	lambda_map: 69 start_index: 116 end_index: 117
63	ExprRange	lambda_map: 70 start_index: 116 end_index: 118
64	ExprTuple	73
65	Lambda	parameter: 110 body: 71
66	Lambda	parameter: 110 body: 72
67	ExprTuple	106
68	ExprTuple	73, 74
69	Lambda	parameter: 110 body: 75
70	Lambda	parameter: 110 body: 76
71	Operation	operator: 82 operands: 77
72	Operation	operator: 82 operands: 78
73	ExprRange	lambda_map: 79 start_index: 116 end_index: 117
74	ExprRange	lambda_map: 80 start_index: 116 end_index: 118
75	Operation	operator: 82 operands: 81
76	Operation	operator: 82 operands: 83
77	NamedExprs	element: 84 targets: 90
78	NamedExprs	element: 85 targets: 92
79	Lambda	parameter: 110 body: 86
80	Lambda	parameter: 110 body: 88
81	NamedExprs	element: 89 targets: 90
82	Literal
83	NamedExprs	element: 91 targets: 92
84	Operation	operator: 94 operands: 93
85	Operation	operator: 94 operands: 102
86	IndexedVar	variable: 95 indices: 96
87	ExprTuple	110
88	Operation	operator: 97 operands: 98
89	Operation	operator: 101 operands: 99
90	Operation	operator: 103 operands: 100
91	Operation	operator: 101 operands: 102
92	Operation	operator: 103 operands: 104
93	NamedExprs	state: 105 part: 110
94	Literal
95	Variable
96	ExprTuple	106, 110
97	Literal
98	NamedExprs	operation: 107
99	NamedExprs	state: 108 part: 110
100	ExprTuple	116, 117
101	Literal
102	NamedExprs	state: 109 part: 110
103	Literal
104	ExprTuple	111, 112
105	Variable
106	Variable
107	Literal
108	Variable
109	Variable
110	Variable
111	Operation	operator: 114 operands: 113
112	Operation	operator: 114 operands: 115
113	ExprTuple	117, 116
114	Literal
115	ExprTuple	117, 118
116	Literal
117	Variable
118	Variable