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 Hspace, OrthoNormBases, OrthoProj, VecSum
from proveit.logic import Equals, Forall, 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(X, Conditional(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), SubsetEq(X, Hspace)))

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())

X \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(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] \textrm{ if } X \subseteq \mathcal{H}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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