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, Dim, HilbertSpaces, Hspace, LinMap, OrthoNormBases, ScalarMult, VecSum
from proveit.logic import Equals, Exists, Forall, Iff, InSet, Set
from proveit.numbers import Complex, Interval, NaturalPos, 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(n, Conditional(Forall(instance_param_or_params = [Hspace], instance_expr = Forall(instance_param_or_params = [M], instance_expr = 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), domain = LinMap(Hspace, Hspace)), domain = HilbertSpaces, condition = Equals(Dim(Hspace), n)).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_{\mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}~|~\textrm{Dim}\left(\mathcal{H}\right) = n}~\\
\left[\forall_{M \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)}~\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}\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: 110 body: 2
1	ExprTuple	110
2	Conditional	value: 3 condition: 4
3	Operation	operator: 12 operand: 7
4	Operation	operator: 95 operands: 6
5	ExprTuple	7
6	ExprTuple	110, 8
7	Lambda	parameter: 66 body: 9
8	Literal
9	Conditional	value: 10 condition: 11
10	Operation	operator: 12 operand: 15
11	Operation	operator: 70 operands: 14
12	Literal
13	ExprTuple	15
14	ExprTuple	16, 17
15	Lambda	parameter: 74 body: 18
16	Operation	operator: 19 operands: 20
17	Operation	operator: 68 operands: 21
18	Conditional	value: 22 condition: 23
19	Literal
20	ExprTuple	66, 24
21	ExprTuple	25, 110
22	Operation	operator: 26 operands: 27
23	Operation	operator: 95 operands: 28
24	Literal
25	Operation	operator: 29 operand: 66
26	Literal
27	ExprTuple	30, 31
28	ExprTuple	74, 32
29	Literal
30	Operation	operator: 68 operands: 33
31	Operation	operator: 51 operand: 39
32	Operation	operator: 35 operands: 36
33	ExprTuple	37, 38
34	ExprTuple	39
35	Literal
36	ExprTuple	66, 66
37	Operation	operator: 103 operands: 40
38	Operation	operator: 103 operands: 41
39	Lambda	parameters: 42 body: 43
40	ExprTuple	44, 74
41	ExprTuple	74, 44
42	ExprTuple	45
43	Conditional	value: 46 condition: 47
44	Operation	operator: 48 operand: 74
45	ExprRange	lambda_map: 50 start_index: 109 end_index: 110
46	Operation	operator: 51 operand: 54
47	Operation	operator: 95 operands: 53
48	Literal
49	ExprTuple	74
50	Lambda	parameter: 101 body: 92
51	Literal
52	ExprTuple	54
53	ExprTuple	55, 56
54	Lambda	parameters: 57 body: 58
55	Operation	operator: 72 operand: 65
56	Operation	operator: 60 operand: 66
57	ExprTuple	62
58	Conditional	value: 63 condition: 64
59	ExprTuple	65
60	Literal
61	ExprTuple	66
62	ExprRange	lambda_map: 67 start_index: 109 end_index: 110
63	Operation	operator: 68 operands: 69
64	Operation	operator: 70 operands: 71
65	Operation	operator: 72 operands: 73
66	Variable
67	Lambda	parameter: 101 body: 90
68	Literal
69	ExprTuple	74, 75
70	Literal
71	ExprTuple	76
72	Literal
73	ExprTuple	77
74	Variable
75	Operation	operator: 78 operand: 82
76	ExprRange	lambda_map: 80 start_index: 109 end_index: 110
77	ExprRange	lambda_map: 81 start_index: 109 end_index: 110
78	Literal
79	ExprTuple	82
80	Lambda	parameter: 101 body: 83
81	Lambda	parameter: 101 body: 84
82	Lambda	parameter: 117 body: 85
83	Operation	operator: 95 operands: 86
84	Operation	operator: 111 operand: 92
85	Conditional	value: 88 condition: 89
86	ExprTuple	90, 91
87	ExprTuple	92
88	Operation	operator: 93 operands: 94
89	Operation	operator: 95 operands: 96
90	IndexedVar	variable: 102 index: 101
91	Literal
92	IndexedVar	variable: 115 index: 101
93	Literal
94	ExprTuple	98, 99
95	Literal
96	ExprTuple	117, 100
97	ExprTuple	101
98	IndexedVar	variable: 102 index: 117
99	Operation	operator: 103 operands: 104
100	Operation	operator: 105 operands: 106
101	Variable
102	Variable
103	Literal
104	ExprTuple	107, 108
105	Literal
106	ExprTuple	109, 110
107	Operation	operator: 111 operand: 114
108	Operation	operator: 112 operand: 114
109	Literal
110	Variable
111	Literal
112	Literal
113	ExprTuple	114
114	IndexedVar	variable: 115 index: 117
115	Variable
116	ExprTuple	117
117	Variable