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, NaturalPos, RealNonNeg
from proveit.physics.quantum import Qmult, var_ket_psi

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

\forall_{n \in \mathbb{N}^+}~\left[\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]\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: 24 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 55 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 24 operand: 8
5	Operation	operator: 39 operands: 7
6	ExprTuple	8
7	ExprTuple	55, 9
8	Lambda	parameter: 53 body: 11
9	Literal
10	ExprTuple	53
11	Conditional	value: 12 condition: 13
12	Operation	operator: 24 operand: 16
13	Operation	operator: 39 operands: 15
14	ExprTuple	16
15	ExprTuple	53, 17
16	Lambda	parameter: 46 body: 19
17	Operation	operator: 20 operand: 55
18	ExprTuple	46
19	Conditional	value: 22 condition: 23
20	Literal
21	ExprTuple	55
22	Operation	operator: 24 operand: 27
23	Operation	operator: 39 operands: 26
24	Literal
25	ExprTuple	27
26	ExprTuple	46, 28
27	Lambda	parameter: 56 body: 29
28	Literal
29	Conditional	value: 30 condition: 31
30	Operation	operator: 41 operands: 32
31	Operation	operator: 33 operands: 34
32	ExprTuple	35, 46
33	Literal
34	ExprTuple	36, 37
35	Operation	operator: 51 operand: 43
36	Operation	operator: 39 operands: 40
37	Operation	operator: 41 operands: 42
38	ExprTuple	43
39	Literal
40	ExprTuple	56, 44
41	Literal
42	ExprTuple	45, 46
43	Operation	operator: 47 operands: 48
44	Operation	operator: 49 operands: 50
45	Operation	operator: 51 operand: 56
46	Variable
47	Literal
48	ExprTuple	53, 56
49	Literal
50	ExprTuple	54, 55
51	Literal
52	ExprTuple	56
53	Variable
54	Literal
55	Variable
56	Variable