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 Adj, Hspace, LinMap, OrthoNormBases, ScalarMult, VecSum
from proveit.logic import Equals, Exists, Iff, 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
sub_expr1 = Adj(M)
expr = Lambda(M, Conditional(Iff(Equals(Qmult(sub_expr1, M), Qmult(M, sub_expr1)), Exists(instance_param_or_params = [v_1_to_n], instance_expr = 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(), condition = InSet(Set(v_1_to_n_kets), OrthoNormBases(Hspace))).with_wrapping()).with_wrapping_at(2), InSet(M, LinMap(Hspace, 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())

M \mapsto \left\{\begin{array}{c} \begin{array}{l} \left(\left(M^{\dagger} \thinspace M\right) = \left(M \thinspace M^{\dagger}\right)\right) \Leftrightarrow  \\ \left[\begin{array}{l}\exists_{v_{1}, v_{2}, \ldots, v_{n}~|~\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)}~\\
\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}\right]\end{array}\right] \end{array} \end{array} \textrm{ if } M \in \mathcal{L}\left(\mathcal{H}, \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	parameter: 51 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operands: 5
3	Operation	operator: 72 operands: 6
4	Literal
5	ExprTuple	7, 8
6	ExprTuple	51, 9
7	Operation	operator: 45 operands: 10
8	Operation	operator: 28 operand: 16
9	Operation	operator: 12 operands: 13
10	ExprTuple	14, 15
11	ExprTuple	16
12	Literal
13	ExprTuple	43, 43
14	Operation	operator: 80 operands: 17
15	Operation	operator: 80 operands: 18
16	Lambda	parameters: 19 body: 20
17	ExprTuple	21, 51
18	ExprTuple	51, 21
19	ExprTuple	22
20	Conditional	value: 23 condition: 24
21	Operation	operator: 25 operand: 51
22	ExprRange	lambda_map: 27 start_index: 86 end_index: 87
23	Operation	operator: 28 operand: 31
24	Operation	operator: 72 operands: 30
25	Literal
26	ExprTuple	51
27	Lambda	parameter: 78 body: 69
28	Literal
29	ExprTuple	31
30	ExprTuple	32, 33
31	Lambda	parameters: 34 body: 35
32	Operation	operator: 49 operand: 42
33	Operation	operator: 37 operand: 43
34	ExprTuple	39
35	Conditional	value: 40 condition: 41
36	ExprTuple	42
37	Literal
38	ExprTuple	43
39	ExprRange	lambda_map: 44 start_index: 86 end_index: 87
40	Operation	operator: 45 operands: 46
41	Operation	operator: 47 operands: 48
42	Operation	operator: 49 operands: 50
43	Variable
44	Lambda	parameter: 78 body: 67
45	Literal
46	ExprTuple	51, 52
47	Literal
48	ExprTuple	53
49	Literal
50	ExprTuple	54
51	Variable
52	Operation	operator: 55 operand: 59
53	ExprRange	lambda_map: 57 start_index: 86 end_index: 87
54	ExprRange	lambda_map: 58 start_index: 86 end_index: 87
55	Literal
56	ExprTuple	59
57	Lambda	parameter: 78 body: 60
58	Lambda	parameter: 78 body: 61
59	Lambda	parameter: 94 body: 62
60	Operation	operator: 72 operands: 63
61	Operation	operator: 88 operand: 69
62	Conditional	value: 65 condition: 66
63	ExprTuple	67, 68
64	ExprTuple	69
65	Operation	operator: 70 operands: 71
66	Operation	operator: 72 operands: 73
67	IndexedVar	variable: 79 index: 78
68	Literal
69	IndexedVar	variable: 92 index: 78
70	Literal
71	ExprTuple	75, 76
72	Literal
73	ExprTuple	94, 77
74	ExprTuple	78
75	IndexedVar	variable: 79 index: 94
76	Operation	operator: 80 operands: 81
77	Operation	operator: 82 operands: 83
78	Variable
79	Variable
80	Literal
81	ExprTuple	84, 85
82	Literal
83	ExprTuple	86, 87
84	Operation	operator: 88 operand: 91
85	Operation	operator: 89 operand: 91
86	Literal
87	Variable
88	Literal
89	Literal
90	ExprTuple	91
91	IndexedVar	variable: 92 index: 94
92	Variable
93	ExprTuple	94
94	Variable