Expression of type Forall

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.core_expr_types.expr_arrays import A11_to_Akl
from proveit.logic import Equals, Forall
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 = 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)

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())

\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]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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