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.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 = [A11_to_Akl], instance_expr = Equals(Prob(circuit__u_Akl_v), Prob(circuit__psi_m__u_Akl_v)).with_wrapping_at(2)).with_wrapping()

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())

\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}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	6
5	Operation	operator: 7 operands: 8
6	ExprRange	lambda_map: 9 start_index: 95 end_index: 32
7	Literal
8	ExprTuple	10, 11
9	Lambda	parameter: 86 body: 49
10	Operation	operator: 13 operand: 15
11	Operation	operator: 13 operand: 16
12	ExprTuple	15
13	Literal
14	ExprTuple	16
15	Operation	operator: 18 operands: 17
16	Operation	operator: 18 operands: 19
17	ExprTuple	20, 21, 22
18	Literal
19	ExprTuple	23, 24, 25
20	ExprTuple	26
21	ExprRange	lambda_map: 27 start_index: 95 end_index: 32
22	ExprTuple	28
23	ExprTuple	29, 30
24	ExprRange	lambda_map: 31 start_index: 95 end_index: 32
25	ExprTuple	33, 34
26	ExprRange	lambda_map: 35 start_index: 95 end_index: 97
27	Lambda	parameter: 86 body: 36
28	ExprRange	lambda_map: 37 start_index: 95 end_index: 97
29	ExprRange	lambda_map: 38 start_index: 95 end_index: 96
30	ExprRange	lambda_map: 39 start_index: 95 end_index: 97
31	Lambda	parameter: 86 body: 41
32	Variable
33	ExprRange	lambda_map: 42 start_index: 95 end_index: 96
34	ExprRange	lambda_map: 43 start_index: 95 end_index: 97
35	Lambda	parameter: 89 body: 44
36	ExprTuple	49
37	Lambda	parameter: 89 body: 45
38	Lambda	parameter: 89 body: 46
39	Lambda	parameter: 89 body: 47
40	ExprTuple	86
41	ExprTuple	48, 49
42	Lambda	parameter: 89 body: 50
43	Lambda	parameter: 89 body: 51
44	Operation	operator: 59 operands: 52
45	Operation	operator: 59 operands: 53
46	Operation	operator: 59 operands: 54
47	Operation	operator: 59 operands: 55
48	ExprRange	lambda_map: 56 start_index: 95 end_index: 96
49	ExprRange	lambda_map: 57 start_index: 95 end_index: 97
50	Operation	operator: 59 operands: 58
51	Operation	operator: 59 operands: 60
52	NamedExprs	element: 63 targets: 61
53	NamedExprs	element: 69 targets: 61
54	NamedExprs	element: 62 targets: 68
55	NamedExprs	element: 63 targets: 70
56	Lambda	parameter: 89 body: 64
57	Lambda	parameter: 89 body: 66
58	NamedExprs	element: 67 targets: 68
59	Literal
60	NamedExprs	element: 69 targets: 70
61	Operation	operator: 82 operands: 71
62	Operation	operator: 72 operands: 78
63	Operation	operator: 72 operands: 73
64	Operation	operator: 74 operands: 75
65	ExprTuple	89
66	IndexedVar	variable: 76 indices: 77
67	Operation	operator: 80 operands: 78
68	Operation	operator: 82 operands: 79
69	Operation	operator: 80 operands: 81
70	Operation	operator: 82 operands: 83
71	ExprTuple	95, 97
72	Literal
73	NamedExprs	state: 84 part: 89
74	Literal
75	NamedExprs	operation: 85
76	Variable
77	ExprTuple	86, 89
78	NamedExprs	state: 87 part: 89
79	ExprTuple	95, 96
80	Literal
81	NamedExprs	state: 88 part: 89
82	Literal
83	ExprTuple	90, 91
84	Variable
85	Literal
86	Variable
87	Variable
88	Variable
89	Variable
90	Operation	operator: 93 operands: 92
91	Operation	operator: 93 operands: 94
92	ExprTuple	96, 95
93	Literal
94	ExprTuple	96, 97
95	Literal
96	Variable
97	Variable