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, 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 = 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))

# 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 \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()

# display the expression information
stored_expr.expr_info()

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