Expression of type Conditional

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, i, j, m
from proveit.core_expr_types import U_1_to_i, V_1_to_j
from proveit.logic import And, Equals, Forall, Implies, InSet
from proveit.numbers import NaturalPos, one
from proveit.physics.quantum import m_ket_domain, var_ket_psi
from proveit.physics.quantum.circuits import circuit_Ui_Vj, circuit_Ui_psi_m, circuit_psi_m_Vj
from proveit.statistics import Prob

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [var_ket_psi], instance_expr = Forall(instance_param_or_params = [U_1_to_i, V_1_to_j], instance_expr = Implies(Equals(Prob(circuit_Ui_psi_m), one), Equals(Prob(circuit_Ui_Vj), Prob(circuit_psi_m_Vj)).with_wrapping_at(2)).with_wrapping_at(2)).with_wrapping(), domain = m_ket_domain).with_wrapping(), And(InSet(i, NaturalPos), InSet(j, NaturalPos), InSet(m, NaturalPos)))

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())

\left\{\begin{array}{l}\forall_{\lvert \psi \rangle \in \mathbb{C}^{2^{m}}}~\\
\left[\begin{array}{l}\forall_{U_{1}, U_{2}, \ldots, U_{i}, V_{1}, V_{2}, \ldots, V_{j}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{\begin{array}{c} \uparrow \\U_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{i} \\ \downarrow\end{array}} & \qout{\lvert \psi \rangle}
} \end{array}\right) = 1\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
& \gate{\begin{array}{c} \uparrow \\U_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\U_{i} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{2} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{j} \\ \downarrow\end{array}} & \qw
} \end{array}\right) =  \\ \textrm{Pr}\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{\lvert \psi \rangle} & \gate{\begin{array}{c} \uparrow \\V_{1} \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\\cdots \\ \downarrow\end{array}} & \gate{\begin{array}{c} \uparrow \\V_{j} \\ \downarrow\end{array}} & { /^{m} } \qw
} \end{array}\right) \end{array} \end{array}\right) \end{array} \end{array}\right)\end{array}\right]\end{array} \textrm{ if } i \in \mathbb{N}^+ ,  j \in \mathbb{N}^+ ,  m \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

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