Expression of type Conditional

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 Conditional, U, alpha, n
from proveit.linear_algebra import Norm, Unitary
from proveit.logic import CartExp, Equals, Forall, InSet
from proveit.numbers import Complex, RealNonNeg
from proveit.physics.quantum import Qmult, var_ket_psi

# build up the expression from sub-expressions
expr = Conditional(Forall(instance_param_or_params = [alpha], instance_expr = Forall(instance_param_or_params = [var_ket_psi], instance_expr = Equals(Norm(Qmult(U, var_ket_psi)), alpha), domain = CartExp(Complex, n), condition = Equals(Norm(var_ket_psi), alpha)), domain = RealNonNeg), InSet(U, Unitary(n)))

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())

\left\{\forall_{\alpha \in \mathbb{R}^{\ge 0}}~\left[\forall_{\lvert \psi \rangle \in \mathbb{C}^{n}~|~\left \|\lvert \psi \rangle\right \| = \alpha}~\left(\left \|U \thinspace \lvert \psi \rangle\right \| = \alpha\right)\right] \textrm{ if } U \in \textrm{U}\left(n\right)\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

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