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, X, i, n
from proveit.core_expr_types import x_1_to_n, x_i
from proveit.linear_algebra import HilbertSpaces, Hspace, OrthoNormBases, OrthoProj, VecSum
from proveit.logic import Equals, Forall, InClass, InSet, SubsetEq
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 = Conditional(Forall(instance_param_or_params = [X], instance_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(X))).with_wrapping(), domain = NaturalPos), condition = SubsetEq(X, Hspace)), InClass(Hspace, HilbertSpaces))

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_{X~|~X \subseteq \mathcal{H}}~\left[\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(X\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]\right] \textrm{ if } \mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}\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: 19 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	45, 7
6	Lambda	parameter: 48 body: 8
7	Literal
8	Conditional	value: 9 condition: 10
9	Operation	operator: 19 operand: 14
10	Operation	operator: 12 operands: 13
11	ExprTuple	14
12	Literal
13	ExprTuple	48, 45
14	Lambda	parameter: 71 body: 16
15	ExprTuple	71
16	Conditional	value: 17 condition: 18
17	Operation	operator: 19 operand: 22
18	Operation	operator: 56 operands: 21
19	Literal
20	ExprTuple	22
21	ExprTuple	71, 23
22	Lambda	parameters: 24 body: 25
23	Literal
24	ExprTuple	26
25	Conditional	value: 27 condition: 28
26	ExprRange	lambda_map: 29 start_index: 70 end_index: 71
27	Operation	operator: 30 operands: 31
28	Operation	operator: 56 operands: 32
29	Lambda	parameter: 72 body: 62
30	Literal
31	ExprTuple	33, 34
32	ExprTuple	35, 36
33	Operation	operator: 37 operands: 38
34	Operation	operator: 39 operand: 46
35	Operation	operator: 41 operands: 42
36	Operation	operator: 43 operand: 48
37	Literal
38	ExprTuple	45, 48
39	Literal
40	ExprTuple	46
41	Literal
42	ExprTuple	47
43	Literal
44	ExprTuple	48
45	Variable
46	Lambda	parameter: 75 body: 49
47	ExprRange	lambda_map: 50 start_index: 70 end_index: 71
48	Variable
49	Conditional	value: 51 condition: 52
50	Lambda	parameter: 72 body: 53
51	Operation	operator: 54 operands: 55
52	Operation	operator: 56 operands: 57
53	Operation	operator: 63 operand: 62
54	Literal
55	ExprTuple	59, 60
56	Literal
57	ExprTuple	75, 61
58	ExprTuple	62
59	Operation	operator: 63 operand: 69
60	Operation	operator: 64 operand: 69
61	Operation	operator: 66 operands: 67
62	IndexedVar	variable: 73 index: 72
63	Literal
64	Literal
65	ExprTuple	69
66	Literal
67	ExprTuple	70, 71
68	ExprTuple	72
69	IndexedVar	variable: 73 index: 75
70	Literal
71	Variable
72	Variable
73	Variable
74	ExprTuple	75
75	Variable