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.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 = [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()

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_{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}

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	28, 30
5	Operation	operator: 6 operands: 7
6	Literal
7	ExprTuple	8, 9
8	Operation	operator: 11 operands: 10
9	Operation	operator: 11 operands: 12
10	ExprTuple	13, 59
11	Literal
12	ExprTuple	14, 15
13	Operation	operator: 18 operand: 20
14	Operation	operator: 18 operand: 21
15	Operation	operator: 18 operand: 22
16	ExprTuple	20
17	ExprTuple	21
18	Literal
19	ExprTuple	22
20	Operation	operator: 25 operands: 23
21	Operation	operator: 25 operands: 24
22	Operation	operator: 25 operands: 26
23	ExprTuple	28, 27
24	ExprTuple	28, 30
25	Literal
26	ExprTuple	29, 30
27	ExprTuple	31
28	ExprRange	lambda_map: 32 start_index: 59 end_index: 33
29	ExprTuple	34
30	ExprRange	lambda_map: 35 start_index: 59 end_index: 36
31	ExprRange	lambda_map: 37 start_index: 59 end_index: 60
32	Lambda	parameter: 58 body: 38
33	Variable
34	ExprRange	lambda_map: 39 start_index: 59 end_index: 60
35	Lambda	parameter: 58 body: 40
36	Variable
37	Lambda	parameter: 58 body: 41
38	IndexedVar	variable: 42 index: 58
39	Lambda	parameter: 58 body: 43
40	IndexedVar	variable: 44 index: 58
41	Operation	operator: 47 operands: 46
42	Variable
43	Operation	operator: 47 operands: 48
44	Variable
45	ExprTuple	58
46	NamedExprs	element: 49 targets: 51
47	Literal
48	NamedExprs	element: 50 targets: 51
49	Operation	operator: 52 operands: 54
50	Operation	operator: 53 operands: 54
51	Operation	operator: 55 operands: 56
52	Literal
53	Literal
54	NamedExprs	state: 57 part: 58
55	Literal
56	ExprTuple	59, 60
57	Variable
58	Variable
59	Literal
60	Variable