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, i, n
from proveit.core_expr_types import x_1_to_n, x_i
from proveit.linear_algebra import Hspace, Identity, OrthoNormBases, 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 = Conditional(Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(VecSum(index_or_indices = [i], summand = Qmult(Ket(x_i), Bra(x_i)), domain = Interval(one, n)), Identity(Hspace)), condition = InSet(x_1_to_n_kets, OrthoNormBases(Hspace))).with_wrapping(), InSet(n, 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())

\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(\left(\sum_{i=1}^{n} \left(\lvert x_{i} \rangle \thinspace \langle x_{i} \rvert\right)\right) = I_{\mathcal{H}}\right)\end{array} \textrm{ if } n \in \mathbb{N}^+\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: 3 operand: 6
2	Operation	operator: 38 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	53, 7
6	Lambda	parameters: 8 body: 9
7	Literal
8	ExprTuple	10
9	Conditional	value: 11 condition: 12
10	ExprRange	lambda_map: 13 start_index: 52 end_index: 53
11	Operation	operator: 14 operands: 15
12	Operation	operator: 38 operands: 16
13	Lambda	parameter: 54 body: 44
14	Literal
15	ExprTuple	17, 18
16	ExprTuple	19, 20
17	Operation	operator: 21 operand: 28
18	Operation	operator: 23 operand: 30
19	Operation	operator: 24 operands: 25
20	Operation	operator: 26 operand: 30
21	Literal
22	ExprTuple	28
23	Literal
24	Literal
25	ExprTuple	29
26	Literal
27	ExprTuple	30
28	Lambda	parameter: 57 body: 31
29	ExprRange	lambda_map: 32 start_index: 52 end_index: 53
30	Variable
31	Conditional	value: 33 condition: 34
32	Lambda	parameter: 54 body: 35
33	Operation	operator: 36 operands: 37
34	Operation	operator: 38 operands: 39
35	Operation	operator: 45 operand: 44
36	Literal
37	ExprTuple	41, 42
38	Literal
39	ExprTuple	57, 43
40	ExprTuple	44
41	Operation	operator: 45 operand: 51
42	Operation	operator: 46 operand: 51
43	Operation	operator: 48 operands: 49
44	IndexedVar	variable: 55 index: 54
45	Literal
46	Literal
47	ExprTuple	51
48	Literal
49	ExprTuple	52, 53
50	ExprTuple	54
51	IndexedVar	variable: 55 index: 57
52	Literal
53	Variable
54	Variable
55	Variable
56	ExprTuple	57
57	Variable