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 X, i, n
from proveit.core_expr_types import x_1_to_n, x_i
from proveit.linear_algebra import Hspace, OrthoNormBases, OrthoProj, VecSum
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Interval, NaturalPos, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import x_1_to_n_kets

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(OrthoProj(Hspace, X), VecSum(index_or_indices = [i], summand = Qmult(Ket(x_i), Bra(x_i)), domain = Interval(one, n))), condition = InSet(x_1_to_n_kets, OrthoNormBases(Hspace))).with_wrapping(), 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[\begin{array}{l}\forall_{x_{1}, x_{2}, \ldots, x_{n}~|~\left\{\lvert x_{1} \rangle, \lvert x_{2} \rangle, \ldots, \lvert x_{n} \rangle\right\} \in \textrm{O.N.Bases}\left(\mathcal{H}\right)}~\\
\left(\textrm{OrthoProj}\left(\mathcal{H}, X\right) = \left(\sum_{i=1}^{n} \left(\lvert x_{i} \rangle \thinspace \langle x_{i} \rvert\right)\right)\right)\end{array}\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 59 body: 4
3	ExprTuple	59
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 44 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	59, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	14
13	Conditional	value: 15 condition: 16
14	ExprRange	lambda_map: 17 start_index: 58 end_index: 59
15	Operation	operator: 18 operands: 19
16	Operation	operator: 44 operands: 20
17	Lambda	parameter: 60 body: 50
18	Literal
19	ExprTuple	21, 22
20	ExprTuple	23, 24
21	Operation	operator: 25 operands: 26
22	Operation	operator: 27 operand: 34
23	Operation	operator: 29 operands: 30
24	Operation	operator: 31 operand: 36
25	Literal
26	ExprTuple	36, 33
27	Literal
28	ExprTuple	34
29	Literal
30	ExprTuple	35
31	Literal
32	ExprTuple	36
33	Variable
34	Lambda	parameter: 63 body: 37
35	ExprRange	lambda_map: 38 start_index: 58 end_index: 59
36	Variable
37	Conditional	value: 39 condition: 40
38	Lambda	parameter: 60 body: 41
39	Operation	operator: 42 operands: 43
40	Operation	operator: 44 operands: 45
41	Operation	operator: 51 operand: 50
42	Literal
43	ExprTuple	47, 48
44	Literal
45	ExprTuple	63, 49
46	ExprTuple	50
47	Operation	operator: 51 operand: 57
48	Operation	operator: 52 operand: 57
49	Operation	operator: 54 operands: 55
50	IndexedVar	variable: 61 index: 60
51	Literal
52	Literal
53	ExprTuple	57
54	Literal
55	ExprTuple	58, 59
56	ExprTuple	60
57	IndexedVar	variable: 61 index: 63
58	Literal
59	Variable
60	Variable
61	Variable
62	ExprTuple	63
63	Variable