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(Hspace))).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(\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]\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: 20 operand: 6
2	Operation	operator: 4 operands: 5
3	ExprTuple	6
4	Literal
5	ExprTuple	49, 7
6	Lambda	parameter: 46 body: 9
7	Literal
8	ExprTuple	46
9	Conditional	value: 10 condition: 11
10	Operation	operator: 20 operand: 15
11	Operation	operator: 13 operands: 14
12	ExprTuple	15
13	Literal
14	ExprTuple	46, 49
15	Lambda	parameter: 72 body: 17
16	ExprTuple	72
17	Conditional	value: 18 condition: 19
18	Operation	operator: 20 operand: 23
19	Operation	operator: 57 operands: 22
20	Literal
21	ExprTuple	23
22	ExprTuple	72, 24
23	Lambda	parameters: 25 body: 26
24	Literal
25	ExprTuple	27
26	Conditional	value: 28 condition: 29
27	ExprRange	lambda_map: 30 start_index: 71 end_index: 72
28	Operation	operator: 31 operands: 32
29	Operation	operator: 57 operands: 33
30	Lambda	parameter: 73 body: 63
31	Literal
32	ExprTuple	34, 35
33	ExprTuple	36, 37
34	Operation	operator: 38 operands: 39
35	Operation	operator: 40 operand: 47
36	Operation	operator: 42 operands: 43
37	Operation	operator: 44 operand: 49
38	Literal
39	ExprTuple	49, 46
40	Literal
41	ExprTuple	47
42	Literal
43	ExprTuple	48
44	Literal
45	ExprTuple	49
46	Variable
47	Lambda	parameter: 76 body: 50
48	ExprRange	lambda_map: 51 start_index: 71 end_index: 72
49	Variable
50	Conditional	value: 52 condition: 53
51	Lambda	parameter: 73 body: 54
52	Operation	operator: 55 operands: 56
53	Operation	operator: 57 operands: 58
54	Operation	operator: 64 operand: 63
55	Literal
56	ExprTuple	60, 61
57	Literal
58	ExprTuple	76, 62
59	ExprTuple	63
60	Operation	operator: 64 operand: 70
61	Operation	operator: 65 operand: 70
62	Operation	operator: 67 operands: 68
63	IndexedVar	variable: 74 index: 73
64	Literal
65	Literal
66	ExprTuple	70
67	Literal
68	ExprTuple	71, 72
69	ExprTuple	73
70	IndexedVar	variable: 74 index: 76
71	Literal
72	Variable
73	Variable
74	Variable
75	ExprTuple	76
76	Variable