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 import U_1_to_i, V_1_to_j
from proveit.logic import Equals, Forall, Implies
from proveit.numbers import 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 = 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()

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_{\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}

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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 73 body: 4
3	ExprTuple	73
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	73, 12
11	Lambda	parameters: 13 body: 14
12	Operation	operator: 15 operands: 16
13	ExprTuple	44, 46
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	21, 22
19	Literal
20	Operation	operator: 23 operands: 24
21	Operation	operator: 26 operands: 25
22	Operation	operator: 26 operands: 27
23	Literal
24	ExprTuple	28, 76
25	ExprTuple	29, 75
26	Literal
27	ExprTuple	30, 31
28	Literal
29	Operation	operator: 34 operand: 36
30	Operation	operator: 34 operand: 37
31	Operation	operator: 34 operand: 38
32	ExprTuple	36
33	ExprTuple	37
34	Literal
35	ExprTuple	38
36	Operation	operator: 41 operands: 39
37	Operation	operator: 41 operands: 40
38	Operation	operator: 41 operands: 42
39	ExprTuple	44, 43
40	ExprTuple	44, 46
41	Literal
42	ExprTuple	45, 46
43	ExprTuple	47
44	ExprRange	lambda_map: 48 start_index: 75 end_index: 49
45	ExprTuple	50
46	ExprRange	lambda_map: 51 start_index: 75 end_index: 52
47	ExprRange	lambda_map: 53 start_index: 75 end_index: 76
48	Lambda	parameter: 74 body: 54
49	Variable
50	ExprRange	lambda_map: 55 start_index: 75 end_index: 76
51	Lambda	parameter: 74 body: 56
52	Variable
53	Lambda	parameter: 74 body: 57
54	IndexedVar	variable: 58 index: 74
55	Lambda	parameter: 74 body: 59
56	IndexedVar	variable: 60 index: 74
57	Operation	operator: 63 operands: 62
58	Variable
59	Operation	operator: 63 operands: 64
60	Variable
61	ExprTuple	74
62	NamedExprs	element: 65 targets: 67
63	Literal
64	NamedExprs	element: 66 targets: 67
65	Operation	operator: 68 operands: 70
66	Operation	operator: 69 operands: 70
67	Operation	operator: 71 operands: 72
68	Literal
69	Literal
70	NamedExprs	state: 73 part: 74
71	Literal
72	ExprTuple	75, 76
73	Variable
74	Variable
75	Literal
76	Variable