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 A, Conditional, m
from proveit.core_expr_types import B_1_to_m
from proveit.linear_algebra import TensorProd
from proveit.logic import Equals, Forall, Iff, InSet
from proveit.numbers import NaturalPos
from proveit.physics.quantum.circuits import QcircuitEquiv, circuit_compressed_inputAm, circuit_expanded_inputBm

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [A, B_1_to_m], instance_expr = Iff(QcircuitEquiv(circuit_compressed_inputAm, circuit_expanded_inputBm), Equals(A, TensorProd(B_1_to_m)))), 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_{1}, B_{2}, \ldots, B_{m}}~\left(\left(\left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{A} & { /^{m} } \qw
} \end{array}\right) \cong \left(\begin{array}{c} \Qcircuit@C=1em @R=.7em{
\qin{B_{1}} & \qw \\
\qin{B_{2}} & \qw \\
\qin{\vdots} & \qw \\
\qin{B_{m}} & \qw
} \end{array}\right)\right) \Leftrightarrow \left(A = \left(B_{1} {\otimes}  B_{2} {\otimes}  \ldots {\otimes}  B_{m}\right)\right)\right) \textrm{ if } 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	51, 8
7	Lambda	parameters: 9 body: 10
8	Literal
9	ExprTuple	49, 29
10	Operation	operator: 11 operands: 12
11	Literal
12	ExprTuple	13, 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19, 20
17	Literal
18	ExprTuple	49, 21
19	Operation	operator: 23 operand: 27
20	Operation	operator: 23 operand: 28
21	Operation	operator: 25 operands: 26
22	ExprTuple	27
23	Literal
24	ExprTuple	28
25	Literal
26	ExprTuple	29
27	ExprTuple	30
28	ExprTuple	31
29	ExprRange	lambda_map: 32 start_index: 50 end_index: 51
30	ExprRange	lambda_map: 33 start_index: 50 end_index: 51
31	ExprRange	lambda_map: 34 start_index: 50 end_index: 51
32	Lambda	parameter: 52 body: 42
33	Lambda	parameter: 52 body: 35
34	Lambda	parameter: 52 body: 36
35	Operation	operator: 37 operands: 38
36	Operation	operator: 43 operands: 39
37	Literal
38	NamedExprs	element: 40 targets: 41
39	NamedExprs	state: 42
40	Operation	operator: 43 operands: 44
41	Operation	operator: 45 operands: 46
42	IndexedVar	variable: 47 index: 52
43	Literal
44	NamedExprs	state: 49 part: 52
45	Literal
46	ExprTuple	50, 51
47	Variable
48	ExprTuple	52
49	Variable
50	Literal
51	Variable
52	Variable