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, M, i, n
from proveit.core_expr_types import a_1_to_n, a_i, v_1_to_n, v_i
from proveit.linear_algebra import Hspace, OrthoNormBases, ScalarMult, VecSum
from proveit.logic import Equals, Exists, InSet, Set
from proveit.numbers import Complex, Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import v_1_to_n_kets

# build up the expression from sub-expressions
expr = Lambda([v_1_to_n], Conditional(Exists(instance_param_or_params = [a_1_to_n], instance_expr = Equals(M, VecSum(index_or_indices = [i], summand = ScalarMult(a_i, Qmult(Ket(v_i), Bra(v_i))), domain = Interval(one, n))), domain = Complex).with_wrapping(), InSet(Set(v_1_to_n_kets), OrthoNormBases(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())

\left(v_{1}, v_{2}, \ldots, v_{n}\right) \mapsto \left\{\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{C}}~\\
\left(M = \left(\sum_{i=1}^{n} \left(a_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right)\end{array} \textrm{ if } \left\{\left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathcal{H}\right)\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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