Expression of type Lambda

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, Lambda, 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 = Lambda(A, 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())

A \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 41 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operand: 7
3	Operation	operator: 18 operands: 6
4	Literal
5	ExprTuple	7
6	ExprTuple	41, 8
7	Lambda	parameter: 25 body: 10
8	Operation	operator: 11 operands: 12
9	ExprTuple	25
10	Conditional	value: 13 condition: 14
11	Literal
12	NamedExprs	field: 15 rows: 27 columns: 27
13	Operation	operator: 16 operands: 17
14	Operation	operator: 18 operands: 19
15	Literal
16	Literal
17	ExprTuple	41, 20
18	Literal
19	ExprTuple	25, 21
20	Operation	operator: 32 operands: 22
21	Operation	operator: 23 operand: 27
22	ExprTuple	25, 26
23	Literal
24	ExprTuple	27
25	Variable
26	Operation	operator: 28 operands: 29
27	Variable
28	Literal
29	ExprTuple	30, 31
30	Operation	operator: 32 operands: 33
31	Operation	operator: 34 operands: 35
32	Literal
33	ExprTuple	36, 41
34	Literal
35	ExprTuple	37, 38
36	Operation	operator: 39 operand: 41
37	Literal
38	Literal
39	Literal
40	ExprTuple	41
41	Variable