Expression of type Conditional

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 import Conditional, a, b, k, m
from proveit.core_expr_types import U_1_to_m
from proveit.logic import And, Forall, Implies, InSet
from proveit.numbers import NaturalPos
from proveit.physics.quantum.circuits import QcircuitEquiv, circuit_aU, circuit_aUb, circuit_b

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [a, b, U_1_to_m], instance_expr = Implies(circuit_aUb, QcircuitEquiv(circuit_aU, circuit_b))), And(InSet(k, NaturalPos), InSet(m, NaturalPos)))

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\{\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) \textrm{ if } k \in \mathbb{N}^+ ,  m \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 7
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7
5	Literal
6	ExprTuple	8, 9
7	Lambda	parameters: 10 body: 11
8	Operation	operator: 13 operands: 12
9	Operation	operator: 13 operands: 14
10	ExprTuple	60, 61, 32
11	Operation	operator: 15 operands: 16
12	ExprTuple	64, 17
13	Literal
14	ExprTuple	37, 17
15	Literal
16	ExprTuple	18, 19
17	Literal
18	Operation	operator: 28 operands: 20
19	Operation	operator: 21 operands: 22
20	ExprTuple	31, 32, 23
21	Literal
22	ExprTuple	24, 25
23	ExprTuple	26
24	Operation	operator: 28 operands: 27
25	Operation	operator: 28 operand: 33
26	ExprRange	lambda_map: 30 start_index: 63 end_index: 64
27	ExprTuple	31, 32
28	Literal
29	ExprTuple	33
30	Lambda	parameter: 62 body: 34
31	ExprTuple	35
32	ExprRange	lambda_map: 36 start_index: 63 end_index: 37
33	ExprTuple	38
34	Operation	operator: 50 operands: 39
35	ExprRange	lambda_map: 40 start_index: 63 end_index: 64
36	Lambda	parameter: 62 body: 41
37	Variable
38	ExprRange	lambda_map: 42 start_index: 63 end_index: 64
39	NamedExprs	element: 43 targets: 54
40	Lambda	parameter: 62 body: 44
41	IndexedVar	variable: 45 index: 62
42	Lambda	parameter: 62 body: 47
43	Operation	operator: 48 operands: 57
44	Operation	operator: 50 operands: 49
45	Variable
46	ExprTuple	62
47	Operation	operator: 50 operands: 51
48	Literal
49	NamedExprs	element: 52 targets: 54
50	Literal
51	NamedExprs	element: 53 targets: 54
52	Operation	operator: 56 operands: 55
53	Operation	operator: 56 operands: 57
54	Operation	operator: 58 operands: 59
55	NamedExprs	state: 60 part: 62
56	Literal
57	NamedExprs	state: 61 part: 62
58	Literal
59	ExprTuple	63, 64
60	Variable
61	Variable
62	Variable
63	Literal
64	Variable