Expression of type Implies

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.logic import Equals, Implies
from proveit.numbers import one
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 = 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)

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}{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}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(2)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 6 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	8, 54
6	Literal
7	ExprTuple	9, 10
8	Operation	operator: 13 operand: 15
9	Operation	operator: 13 operand: 16
10	Operation	operator: 13 operand: 17
11	ExprTuple	15
12	ExprTuple	16
13	Literal
14	ExprTuple	17
15	Operation	operator: 20 operands: 18
16	Operation	operator: 20 operands: 19
17	Operation	operator: 20 operands: 21
18	ExprTuple	23, 22
19	ExprTuple	23, 25
20	Literal
21	ExprTuple	24, 25
22	ExprTuple	26
23	ExprRange	lambda_map: 27 start_index: 54 end_index: 28
24	ExprTuple	29
25	ExprRange	lambda_map: 30 start_index: 54 end_index: 31
26	ExprRange	lambda_map: 32 start_index: 54 end_index: 55
27	Lambda	parameter: 53 body: 33
28	Variable
29	ExprRange	lambda_map: 34 start_index: 54 end_index: 55
30	Lambda	parameter: 53 body: 35
31	Variable
32	Lambda	parameter: 53 body: 36
33	IndexedVar	variable: 37 index: 53
34	Lambda	parameter: 53 body: 38
35	IndexedVar	variable: 39 index: 53
36	Operation	operator: 42 operands: 41
37	Variable
38	Operation	operator: 42 operands: 43
39	Variable
40	ExprTuple	53
41	NamedExprs	element: 44 targets: 46
42	Literal
43	NamedExprs	element: 45 targets: 46
44	Operation	operator: 47 operands: 49
45	Operation	operator: 48 operands: 49
46	Operation	operator: 50 operands: 51
47	Literal
48	Literal
49	NamedExprs	state: 52 part: 53
50	Literal
51	ExprTuple	54, 55
52	Variable
53	Variable
54	Literal
55	Variable