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 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 = 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)

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 \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]

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