Expression of type Conditional ¶

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, U
from proveit.linear_algebra import Adj, Hspace, Identity, LinMap
from proveit.logic import Equals, Exists, InSet
from proveit.numbers import sqrt

# build up the expression from sub-expressions
sub_expr1 = LinMap(Hspace, Hspace)
expr = Conditional(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(), InSet(A, sub_expr1))

# 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\{\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} \textrm{ if } A \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)\right..

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions
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	47, 22
6	Lambda	parameter: 44 body: 7
7	Conditional	value: 8 condition: 9
8	Operation	operator: 19 operands: 10
9	Operation	operator: 11 operands: 12
10	ExprTuple	47, 13
11	Literal
12	ExprTuple	14, 15
13	Operation	operator: 36 operands: 16
14	Operation	operator: 17 operands: 18
15	Operation	operator: 19 operands: 20
16	ExprTuple	21, 44
17	Literal
18	ExprTuple	44, 22
19	Literal
20	ExprTuple	23, 24
21	Operation	operator: 25 operands: 26
22	Operation	operator: 27 operands: 28
23	Operation	operator: 36 operands: 29
24	Operation	operator: 30 operand: 35
25	Literal
26	ExprTuple	32, 33
27	Literal
28	ExprTuple	35, 35
29	ExprTuple	34, 44
30	Literal
31	ExprTuple	35
32	Operation	operator: 36 operands: 37
33	Operation	operator: 38 operands: 39
34	Operation	operator: 45 operand: 44
35	Variable
36	Literal
37	ExprTuple	47, 41
38	Literal
39	ExprTuple	42, 43
40	ExprTuple	44
41	Operation	operator: 45 operand: 47
42	Literal
43	Literal
44	Variable
45	Literal
46	ExprTuple	47
47	Variable

Expression of type Conditional¶

from the theory of proveit.linear_algebra.inner_products¶

Expression of type Conditional ¶

from the theory of proveit.linear_algebra.inner_products ¶