Expression of type Conditional

from the theory of proveit.physics.quantum.algebra

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, U, n
from proveit.linear_algebra import Adj, MatrixSpace, Unitary
from proveit.logic import Equals, Exists, InSet
from proveit.numbers import Complex, sqrt
from proveit.physics.quantum import Qmult

# build up the expression from sub-expressions
expr = Conditional(Exists(instance_param_or_params = [U], instance_expr = Equals(A, Qmult(U, sqrt(Qmult(Adj(A), A)))), domain = Unitary(n)), InSet(A, MatrixSpace(field = Complex, rows = n, columns = n)))

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\{\exists_{U \in \textrm{U}\left(n\right)}~\left(A = \left(U \thinspace \sqrt{\left(A^{\dagger} \thinspace A\right)}\right)\right) \textrm{ if } A \in \mathbb{C}^{n \times 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: 6
2	Operation	operator: 17 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	40, 7
6	Lambda	parameter: 24 body: 9
7	Operation	operator: 10 operands: 11
8	ExprTuple	24
9	Conditional	value: 12 condition: 13
10	Literal
11	NamedExprs	field: 14 rows: 26 columns: 26
12	Operation	operator: 15 operands: 16
13	Operation	operator: 17 operands: 18
14	Literal
15	Literal
16	ExprTuple	40, 19
17	Literal
18	ExprTuple	24, 20
19	Operation	operator: 31 operands: 21
20	Operation	operator: 22 operand: 26
21	ExprTuple	24, 25
22	Literal
23	ExprTuple	26
24	Variable
25	Operation	operator: 27 operands: 28
26	Variable
27	Literal
28	ExprTuple	29, 30
29	Operation	operator: 31 operands: 32
30	Operation	operator: 33 operands: 34
31	Literal
32	ExprTuple	35, 40
33	Literal
34	ExprTuple	36, 37
35	Operation	operator: 38 operand: 40
36	Literal
37	Literal
38	Literal
39	ExprTuple	40
40	Variable