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__u_Akl_v, circuit__u_Akl_v__psi_m
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__u_Akl_v__psi_m)).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{
\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]\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	106, 109
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: 110 body: 14
11	Operation	operator: 22 operands: 15
12	Operation	operator: 22 operands: 16
13	ExprTuple	110
14	Conditional	value: 17 condition: 18
15	ExprTuple	106, 19
16	ExprTuple	109, 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	110, 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, 118
33	ExprRange	lambda_map: 38 start_index: 117 end_index: 62
34	Literal
35	ExprTuple	39, 40
36	Literal
37	Operation	operator: 41 operands: 42
38	Lambda	parameter: 107 body: 74
39	Operation	operator: 44 operand: 47
40	Operation	operator: 44 operand: 48
41	Literal
42	ExprTuple	46, 119
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	59
53	ExprRange	lambda_map: 58 start_index: 117 end_index: 62
54	ExprTuple	63
55	ExprTuple	59, 60
56	ExprRange	lambda_map: 61 start_index: 117 end_index: 62
57	ExprTuple	63, 64
58	Lambda	parameter: 107 body: 65
59	ExprRange	lambda_map: 66 start_index: 117 end_index: 118
60	ExprRange	lambda_map: 67 start_index: 117 end_index: 119
61	Lambda	parameter: 107 body: 69
62	Variable
63	ExprRange	lambda_map: 70 start_index: 117 end_index: 118
64	ExprRange	lambda_map: 71 start_index: 117 end_index: 119
65	ExprTuple	74
66	Lambda	parameter: 111 body: 72
67	Lambda	parameter: 111 body: 73
68	ExprTuple	107
69	ExprTuple	74, 75
70	Lambda	parameter: 111 body: 76
71	Lambda	parameter: 111 body: 77
72	Operation	operator: 83 operands: 78
73	Operation	operator: 83 operands: 79
74	ExprRange	lambda_map: 80 start_index: 117 end_index: 118
75	ExprRange	lambda_map: 81 start_index: 117 end_index: 119
76	Operation	operator: 83 operands: 82
77	Operation	operator: 83 operands: 84
78	NamedExprs	element: 85 targets: 91
79	NamedExprs	element: 86 targets: 93
80	Lambda	parameter: 111 body: 87
81	Lambda	parameter: 111 body: 89
82	NamedExprs	element: 90 targets: 91
83	Literal
84	NamedExprs	element: 92 targets: 93
85	Operation	operator: 95 operands: 94
86	Operation	operator: 95 operands: 103
87	IndexedVar	variable: 96 indices: 97
88	ExprTuple	111
89	Operation	operator: 98 operands: 99
90	Operation	operator: 102 operands: 100
91	Operation	operator: 104 operands: 101
92	Operation	operator: 102 operands: 103
93	Operation	operator: 104 operands: 105
94	NamedExprs	state: 106 part: 111
95	Literal
96	Variable
97	ExprTuple	107, 111
98	Literal
99	NamedExprs	operation: 108
100	NamedExprs	state: 109 part: 111
101	ExprTuple	117, 118
102	Literal
103	NamedExprs	state: 110 part: 111
104	Literal
105	ExprTuple	112, 113
106	Variable
107	Variable
108	Literal
109	Variable
110	Variable
111	Variable
112	Operation	operator: 115 operands: 114
113	Operation	operator: 115 operands: 116
114	ExprTuple	118, 117
115	Literal
116	ExprTuple	118, 119
117	Literal
118	Variable
119	Variable