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
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(n, Conditional(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(), 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())

n \mapsto \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} \textrm{ if } n \in \mathbb{N}^+\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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