Expression of type Forall

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

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [U], instance_expr = 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), domain = 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())

\forall_{U \in \textrm{U}\left(n\right)}~\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]\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

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