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, 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 = Lambda(Hspace, 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())

\mathcal{H} \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 50 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 21 operand: 7
3	Operation	operator: 5 operands: 6
4	ExprTuple	7
5	Literal
6	ExprTuple	50, 8
7	Lambda	parameter: 47 body: 10
8	Literal
9	ExprTuple	47
10	Conditional	value: 11 condition: 12
11	Operation	operator: 21 operand: 16
12	Operation	operator: 14 operands: 15
13	ExprTuple	16
14	Literal
15	ExprTuple	47, 50
16	Lambda	parameter: 73 body: 18
17	ExprTuple	73
18	Conditional	value: 19 condition: 20
19	Operation	operator: 21 operand: 24
20	Operation	operator: 58 operands: 23
21	Literal
22	ExprTuple	24
23	ExprTuple	73, 25
24	Lambda	parameters: 26 body: 27
25	Literal
26	ExprTuple	28
27	Conditional	value: 29 condition: 30
28	ExprRange	lambda_map: 31 start_index: 72 end_index: 73
29	Operation	operator: 32 operands: 33
30	Operation	operator: 58 operands: 34
31	Lambda	parameter: 74 body: 64
32	Literal
33	ExprTuple	35, 36
34	ExprTuple	37, 38
35	Operation	operator: 39 operands: 40
36	Operation	operator: 41 operand: 48
37	Operation	operator: 43 operands: 44
38	Operation	operator: 45 operand: 50
39	Literal
40	ExprTuple	50, 47
41	Literal
42	ExprTuple	48
43	Literal
44	ExprTuple	49
45	Literal
46	ExprTuple	50
47	Variable
48	Lambda	parameter: 77 body: 51
49	ExprRange	lambda_map: 52 start_index: 72 end_index: 73
50	Variable
51	Conditional	value: 53 condition: 54
52	Lambda	parameter: 74 body: 55
53	Operation	operator: 56 operands: 57
54	Operation	operator: 58 operands: 59
55	Operation	operator: 65 operand: 64
56	Literal
57	ExprTuple	61, 62
58	Literal
59	ExprTuple	77, 63
60	ExprTuple	64
61	Operation	operator: 65 operand: 71
62	Operation	operator: 66 operand: 71
63	Operation	operator: 68 operands: 69
64	IndexedVar	variable: 75 index: 74
65	Literal
66	Literal
67	ExprTuple	71
68	Literal
69	ExprTuple	72, 73
70	ExprTuple	74
71	IndexedVar	variable: 75 index: 77
72	Literal
73	Variable
74	Variable
75	Variable
76	ExprTuple	77
77	Variable