Expression of type Lambda

from the theory of proveit.linear_algebra.inner_products

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, Composition, Conditional, Lambda, U
from proveit.linear_algebra import Adj, Hspace, Identity, LinMap
from proveit.logic import And, Equals, InSet
from proveit.numbers import sqrt

# build up the expression from sub-expressions
expr = Lambda(U, Conditional(Equals(A, Composition(sqrt(Composition(A, Adj(A))), U)), And(InSet(U, LinMap(Hspace, Hspace)), Equals(Composition(Adj(U), U), Identity(Hspace)))))

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())

U \mapsto \left\{A = \left(\sqrt{\left(A \circ A^{\dagger}\right)} \circ U\right) \textrm{ if } U \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right) ,  \left(U^{\dagger} \circ U\right) = I_{\mathcal{H}}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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