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, HilbertSpaces, Hspace, Identity, LinMap
from proveit.logic import Equals, Exists, Forall, InClass
from proveit.numbers import sqrt

# build up the expression from sub-expressions
sub_expr1 = LinMap(Hspace, Hspace)
expr = Lambda(Hspace, Conditional(Forall(instance_param_or_params = [A], instance_expr = Exists(instance_param_or_params = [U], instance_expr = Equals(A, Composition(sqrt(Composition(A, Adj(A))), U)), domain = sub_expr1, condition = Equals(Composition(Adj(U), U), Identity(Hspace))).with_wrapping(), domain = sub_expr1), InClass(Hspace, HilbertSpaces)))

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())

\mathcal{H} \mapsto \left\{\forall_{A \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)}~\left[\begin{array}{l}\exists_{U \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)~|~\left(U^{\dagger} \circ U\right) = I_{\mathcal{H}}}~\\
\left(A = \left(\sqrt{\left(A \circ A^{\dagger}\right)} \circ U\right)\right)\end{array}\right] \textrm{ if } \mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 45 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operand: 8
3	Operation	operator: 6 operands: 7
4	Literal
5	ExprTuple	8
6	Literal
7	ExprTuple	45, 9
8	Lambda	parameter: 57 body: 10
9	Literal
10	Conditional	value: 11 condition: 12
11	Operation	operator: 13 operand: 16
12	Operation	operator: 27 operands: 15
13	Literal
14	ExprTuple	16
15	ExprTuple	57, 32
16	Lambda	parameter: 54 body: 17
17	Conditional	value: 18 condition: 19
18	Operation	operator: 29 operands: 20
19	Operation	operator: 21 operands: 22
20	ExprTuple	57, 23
21	Literal
22	ExprTuple	24, 25
23	Operation	operator: 46 operands: 26
24	Operation	operator: 27 operands: 28
25	Operation	operator: 29 operands: 30
26	ExprTuple	31, 54
27	Literal
28	ExprTuple	54, 32
29	Literal
30	ExprTuple	33, 34
31	Operation	operator: 35 operands: 36
32	Operation	operator: 37 operands: 38
33	Operation	operator: 46 operands: 39
34	Operation	operator: 40 operand: 45
35	Literal
36	ExprTuple	42, 43
37	Literal
38	ExprTuple	45, 45
39	ExprTuple	44, 54
40	Literal
41	ExprTuple	45
42	Operation	operator: 46 operands: 47
43	Operation	operator: 48 operands: 49
44	Operation	operator: 55 operand: 54
45	Variable
46	Literal
47	ExprTuple	57, 51
48	Literal
49	ExprTuple	52, 53
50	ExprTuple	54
51	Operation	operator: 55 operand: 57
52	Literal
53	Literal
54	Variable
55	Literal
56	ExprTuple	57
57	Variable