Theorem circuit_output_equiv of type Forall

from the theory of proveit.physics.quantum.circuits

see dependencies

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.physics.quantum.circuits import circuit_output_equiv

# check that the built expression is the same as the stored expression
assert circuit_output_equiv.expr == stored_expr
assert circuit_output_equiv.expr._style_id == stored_expr._style_id
print("Passed sanity check: circuit_output_equiv matches stored_expr")

Passed sanity check: circuit_output_equiv matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{k, m \in \mathbb{N}^+}~\left[\forall_{a, b, U_{1}, U_{2}, \ldots, U_{m}}~\left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{a} & \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_{m} \\ \downarrow\end{array}} & \qout{b}
} \end{array}\right) \Rightarrow \left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{a} & \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_{m} \\ \downarrow\end{array}} & { /^{k} } \qw
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{b} & { /^{k} } \qw
} \end{array}\right)\right)\right)\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('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	parameters: 3 body: 4
3	ExprTuple	68, 41
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	12, 13
11	Lambda	parameters: 14 body: 15
12	Operation	operator: 17 operands: 16
13	Operation	operator: 17 operands: 18
14	ExprTuple	64, 65, 36
15	Operation	operator: 19 operands: 20
16	ExprTuple	68, 21
17	Literal
18	ExprTuple	41, 21
19	Literal
20	ExprTuple	22, 23
21	Literal
22	Operation	operator: 32 operands: 24
23	Operation	operator: 25 operands: 26
24	ExprTuple	35, 36, 27
25	Literal
26	ExprTuple	28, 29
27	ExprTuple	30
28	Operation	operator: 32 operands: 31
29	Operation	operator: 32 operand: 37
30	ExprRange	lambda_map: 34 start_index: 67 end_index: 68
31	ExprTuple	35, 36
32	Literal
33	ExprTuple	37
34	Lambda	parameter: 66 body: 38
35	ExprTuple	39
36	ExprRange	lambda_map: 40 start_index: 67 end_index: 41
37	ExprTuple	42
38	Operation	operator: 54 operands: 43
39	ExprRange	lambda_map: 44 start_index: 67 end_index: 68
40	Lambda	parameter: 66 body: 45
41	Variable
42	ExprRange	lambda_map: 46 start_index: 67 end_index: 68
43	NamedExprs	element: 47 targets: 58
44	Lambda	parameter: 66 body: 48
45	IndexedVar	variable: 49 index: 66
46	Lambda	parameter: 66 body: 51
47	Operation	operator: 52 operands: 61
48	Operation	operator: 54 operands: 53
49	Variable
50	ExprTuple	66
51	Operation	operator: 54 operands: 55
52	Literal
53	NamedExprs	element: 56 targets: 58
54	Literal
55	NamedExprs	element: 57 targets: 58
56	Operation	operator: 60 operands: 59
57	Operation	operator: 60 operands: 61
58	Operation	operator: 62 operands: 63
59	NamedExprs	state: 64 part: 66
60	Literal
61	NamedExprs	state: 65 part: 66
62	Literal
63	ExprTuple	67, 68
64	Variable
65	Variable
66	Variable
67	Literal
68	Variable