Expression of type Lambda

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, Lambda
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 = Lambda(var_ket_psi, 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())

\lvert \psi \rangle \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

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