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