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, Forall, InSet
from proveit.numbers import Complex, NaturalPos, sqrt
from proveit.physics.quantum import Qmult

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [A], instance_expr = Exists(instance_param_or_params = [U], instance_expr = Equals(A, Qmult(U, sqrt(Qmult(Adj(A), A)))), domain = Unitary(n)), domain = MatrixSpace(field = Complex, rows = n, columns = n)), InSet(n, 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 \in \mathbb{C}^{n \times n}}~\left[\exists_{U \in \textrm{U}\left(n\right)}~\left(A = \left(U \thinspace \sqrt{\left(A^{\dagger} \thinspace A\right)}\right)\right)\right] \textrm{ if } n \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: 6
2	Operation	operator: 25 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	34, 7
6	Lambda	parameter: 48 body: 8
7	Literal
8	Conditional	value: 9 condition: 10
9	Operation	operator: 11 operand: 14
10	Operation	operator: 25 operands: 13
11	Literal
12	ExprTuple	14
13	ExprTuple	48, 15
14	Lambda	parameter: 32 body: 17
15	Operation	operator: 18 operands: 19
16	ExprTuple	32
17	Conditional	value: 20 condition: 21
18	Literal
19	NamedExprs	field: 22 rows: 34 columns: 34
20	Operation	operator: 23 operands: 24
21	Operation	operator: 25 operands: 26
22	Literal
23	Literal
24	ExprTuple	48, 27
25	Literal
26	ExprTuple	32, 28
27	Operation	operator: 39 operands: 29
28	Operation	operator: 30 operand: 34
29	ExprTuple	32, 33
30	Literal
31	ExprTuple	34
32	Variable
33	Operation	operator: 35 operands: 36
34	Variable
35	Literal
36	ExprTuple	37, 38
37	Operation	operator: 39 operands: 40
38	Operation	operator: 41 operands: 42
39	Literal
40	ExprTuple	43, 48
41	Literal
42	ExprTuple	44, 45
43	Operation	operator: 46 operand: 48
44	Literal
45	Literal
46	Literal
47	ExprTuple	48
48	Variable