Expression of type Lambda

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, Lambda, i, n
from proveit.core_expr_types import x_1_to_n, x_i
from proveit.linear_algebra import HilbertSpaces, Hspace, Identity, OrthoNormBases, VecSum
from proveit.logic import Equals, Forall, InClass, 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 = Lambda(Hspace, Conditional(Forall(instance_param_or_params = [n], instance_expr = 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(), domain = NaturalPos), 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())

\mathcal{H} \mapsto \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(\left(\sum_{i=1}^{n} \left(\lvert x_{i} \rangle \thinspace \langle x_{i} \rvert\right)\right) = I_{\mathcal{H}}\right)\end{array}\right] \textrm{ if } \mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 40 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 13 operand: 7
3	Operation	operator: 5 operands: 6
4	ExprTuple	7
5	Literal
6	ExprTuple	40, 8
7	Lambda	parameter: 63 body: 10
8	Literal
9	ExprTuple	63
10	Conditional	value: 11 condition: 12
11	Operation	operator: 13 operand: 16
12	Operation	operator: 48 operands: 15
13	Literal
14	ExprTuple	16
15	ExprTuple	63, 17
16	Lambda	parameters: 18 body: 19
17	Literal
18	ExprTuple	20
19	Conditional	value: 21 condition: 22
20	ExprRange	lambda_map: 23 start_index: 62 end_index: 63
21	Operation	operator: 24 operands: 25
22	Operation	operator: 48 operands: 26
23	Lambda	parameter: 64 body: 54
24	Literal
25	ExprTuple	27, 28
26	ExprTuple	29, 30
27	Operation	operator: 31 operand: 38
28	Operation	operator: 33 operand: 40
29	Operation	operator: 34 operands: 35
30	Operation	operator: 36 operand: 40
31	Literal
32	ExprTuple	38
33	Literal
34	Literal
35	ExprTuple	39
36	Literal
37	ExprTuple	40
38	Lambda	parameter: 67 body: 41
39	ExprRange	lambda_map: 42 start_index: 62 end_index: 63
40	Variable
41	Conditional	value: 43 condition: 44
42	Lambda	parameter: 64 body: 45
43	Operation	operator: 46 operands: 47
44	Operation	operator: 48 operands: 49
45	Operation	operator: 55 operand: 54
46	Literal
47	ExprTuple	51, 52
48	Literal
49	ExprTuple	67, 53
50	ExprTuple	54
51	Operation	operator: 55 operand: 61
52	Operation	operator: 56 operand: 61
53	Operation	operator: 58 operands: 59
54	IndexedVar	variable: 65 index: 64
55	Literal
56	Literal
57	ExprTuple	61
58	Literal
59	ExprTuple	62, 63
60	ExprTuple	64
61	IndexedVar	variable: 65 index: 67
62	Literal
63	Variable
64	Variable
65	Variable
66	ExprTuple	67
67	Variable