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 A, B, Conditional, Lambda, i, n
from proveit.core_expr_types import a_1_to_n, a_i, b_1_to_n, b_i, v_1_to_n, v_i
from proveit.linear_algebra import Hspace, OrthoNormBases, ScalarMult, VecSum
from proveit.logic import And, Equals, Exists, InSet, Set
from proveit.numbers import Complex, Interval, one
from proveit.physics.quantum import Bra, Ket, Qmult

# build up the expression from sub-expressions
sub_expr1 = [i]
sub_expr2 = Interval(one, n)
sub_expr3 = Qmult(Ket(v_i), Bra(v_i))
expr = Lambda([v_1_to_n], Conditional(Exists(instance_param_or_params = [a_1_to_n, b_1_to_n], instance_expr = And(Equals(A, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(a_i, sub_expr3), domain = sub_expr2)), Equals(B, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(b_i, sub_expr3), domain = sub_expr2))).with_wrapping_at(2), domain = Complex).with_wrapping(), InSet(Set(v_1_to_n), 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}, b_{1}, b_{2}, \ldots, b_{n} \in \mathbb{C}}~\\
\left(\begin{array}{c} \left(A = \left(\sum_{i=1}^{n} \left(a_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right) \land  \\ \left(B = \left(\sum_{i=1}^{n} \left(b_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right) \end{array}\right)\end{array} \textrm{ if } \left\{v_{1}, v_{2}, \ldots, v_{n}\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: 13 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operand: 7
3	Operation	operator: 64 operands: 6
4	Literal
5	ExprTuple	7
6	ExprTuple	8, 9
7	Lambda	parameters: 10 body: 11
8	Operation	operator: 12 operands: 13
9	Operation	operator: 14 operand: 21
10	ExprTuple	16, 17
11	Conditional	value: 18 condition: 19
12	Literal
13	ExprTuple	20
14	Literal
15	ExprTuple	21
16	ExprRange	lambda_map: 22 start_index: 78 end_index: 79
17	ExprRange	lambda_map: 23 start_index: 78 end_index: 79
18	Operation	operator: 25 operands: 24
19	Operation	operator: 25 operands: 26
20	ExprRange	lambda_map: 27 start_index: 78 end_index: 79
21	Variable
22	Lambda	parameter: 60 body: 51
23	Lambda	parameter: 60 body: 52
24	ExprTuple	28, 29
25	Literal
26	ExprTuple	30, 31
27	Lambda	parameter: 60 body: 32
28	Operation	operator: 34 operands: 33
29	Operation	operator: 34 operands: 35
30	ExprRange	lambda_map: 36 start_index: 78 end_index: 79
31	ExprRange	lambda_map: 37 start_index: 78 end_index: 79
32	IndexedVar	variable: 84 index: 60
33	ExprTuple	38, 39
34	Literal
35	ExprTuple	40, 41
36	Lambda	parameter: 60 body: 42
37	Lambda	parameter: 60 body: 43
38	Variable
39	Operation	operator: 45 operand: 49
40	Variable
41	Operation	operator: 45 operand: 50
42	Operation	operator: 64 operands: 47
43	Operation	operator: 64 operands: 48
44	ExprTuple	49
45	Literal
46	ExprTuple	50
47	ExprTuple	51, 53
48	ExprTuple	52, 53
49	Lambda	parameter: 86 body: 54
50	Lambda	parameter: 86 body: 55
51	IndexedVar	variable: 70 index: 60
52	IndexedVar	variable: 71 index: 60
53	Literal
54	Conditional	value: 57 condition: 59
55	Conditional	value: 58 condition: 59
56	ExprTuple	60
57	Operation	operator: 62 operands: 61
58	Operation	operator: 62 operands: 63
59	Operation	operator: 64 operands: 65
60	Variable
61	ExprTuple	66, 68
62	Literal
63	ExprTuple	67, 68
64	Literal
65	ExprTuple	86, 69
66	IndexedVar	variable: 70 index: 86
67	IndexedVar	variable: 71 index: 86
68	Operation	operator: 72 operands: 73
69	Operation	operator: 74 operands: 75
70	Variable
71	Variable
72	Literal
73	ExprTuple	76, 77
74	Literal
75	ExprTuple	78, 79
76	Operation	operator: 80 operand: 83
77	Operation	operator: 81 operand: 83
78	Literal
79	Variable
80	Literal
81	Literal
82	ExprTuple	83
83	IndexedVar	variable: 84 index: 86
84	Variable
85	ExprTuple	86
86	Variable