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 Implies
from proveit.physics.quantum.circuits import QcircuitEquiv, circuit_aU, circuit_aUb, circuit_b

# build up the expression from sub-expressions
expr = Implies(circuit_aUb, QcircuitEquiv(circuit_aU, circuit_b))

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}{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)

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