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, Forall, InSet
from proveit.numbers import Complex, NaturalPos, sqrt
from proveit.physics.quantum import Qmult

# build up the expression from sub-expressions
expr = Lambda(n, Conditional(Forall(instance_param_or_params = [A], instance_expr = Exists(instance_param_or_params = [U], instance_expr = Equals(A, Qmult(U, sqrt(Qmult(Adj(A), A)))), domain = Unitary(n)), domain = MatrixSpace(field = Complex, rows = n, columns = n)), InSet(n, NaturalPos)))

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())

n \mapsto \left\{\forall_{A \in \mathbb{C}^{n \times n}}~\left[\exists_{U \in \textrm{U}\left(n\right)}~\left(A = \left(U \thinspace \sqrt{\left(A^{\dagger} \thinspace A\right)}\right)\right)\right] \textrm{ if } n \in \mathbb{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: 35 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operand: 7
3	Operation	operator: 26 operands: 6
4	Literal
5	ExprTuple	7
6	ExprTuple	35, 8
7	Lambda	parameter: 49 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	49, 16
15	Lambda	parameter: 33 body: 18
16	Operation	operator: 19 operands: 20
17	ExprTuple	33
18	Conditional	value: 21 condition: 22
19	Literal
20	NamedExprs	field: 23 rows: 35 columns: 35
21	Operation	operator: 24 operands: 25
22	Operation	operator: 26 operands: 27
23	Literal
24	Literal
25	ExprTuple	49, 28
26	Literal
27	ExprTuple	33, 29
28	Operation	operator: 40 operands: 30
29	Operation	operator: 31 operand: 35
30	ExprTuple	33, 34
31	Literal
32	ExprTuple	35
33	Variable
34	Operation	operator: 36 operands: 37
35	Variable
36	Literal
37	ExprTuple	38, 39
38	Operation	operator: 40 operands: 41
39	Operation	operator: 42 operands: 43
40	Literal
41	ExprTuple	44, 49
42	Literal
43	ExprTuple	45, 46
44	Operation	operator: 47 operand: 49
45	Literal
46	Literal
47	Literal
48	ExprTuple	49
49	Variable