Expression of type Conditional

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, 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 = 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())

\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()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 10 operand: 5
2	Operation	operator: 93 operands: 4
3	ExprTuple	5
4	ExprTuple	108, 6
5	Lambda	parameter: 64 body: 7
6	Literal
7	Conditional	value: 8 condition: 9
8	Operation	operator: 10 operand: 13
9	Operation	operator: 68 operands: 12
10	Literal
11	ExprTuple	13
12	ExprTuple	14, 15
13	Lambda	parameter: 72 body: 16
14	Operation	operator: 17 operands: 18
15	Operation	operator: 66 operands: 19
16	Conditional	value: 20 condition: 21
17	Literal
18	ExprTuple	64, 22
19	ExprTuple	23, 108
20	Operation	operator: 24 operands: 25
21	Operation	operator: 93 operands: 26
22	Literal
23	Operation	operator: 27 operand: 64
24	Literal
25	ExprTuple	28, 29
26	ExprTuple	72, 30
27	Literal
28	Operation	operator: 66 operands: 31
29	Operation	operator: 49 operand: 37
30	Operation	operator: 33 operands: 34
31	ExprTuple	35, 36
32	ExprTuple	37
33	Literal
34	ExprTuple	64, 64
35	Operation	operator: 101 operands: 38
36	Operation	operator: 101 operands: 39
37	Lambda	parameters: 40 body: 41
38	ExprTuple	42, 72
39	ExprTuple	72, 42
40	ExprTuple	43
41	Conditional	value: 44 condition: 45
42	Operation	operator: 46 operand: 72
43	ExprRange	lambda_map: 48 start_index: 107 end_index: 108
44	Operation	operator: 49 operand: 52
45	Operation	operator: 93 operands: 51
46	Literal
47	ExprTuple	72
48	Lambda	parameter: 99 body: 90
49	Literal
50	ExprTuple	52
51	ExprTuple	53, 54
52	Lambda	parameters: 55 body: 56
53	Operation	operator: 70 operand: 63
54	Operation	operator: 58 operand: 64
55	ExprTuple	60
56	Conditional	value: 61 condition: 62
57	ExprTuple	63
58	Literal
59	ExprTuple	64
60	ExprRange	lambda_map: 65 start_index: 107 end_index: 108
61	Operation	operator: 66 operands: 67
62	Operation	operator: 68 operands: 69
63	Operation	operator: 70 operands: 71
64	Variable
65	Lambda	parameter: 99 body: 88
66	Literal
67	ExprTuple	72, 73
68	Literal
69	ExprTuple	74
70	Literal
71	ExprTuple	75
72	Variable
73	Operation	operator: 76 operand: 80
74	ExprRange	lambda_map: 78 start_index: 107 end_index: 108
75	ExprRange	lambda_map: 79 start_index: 107 end_index: 108
76	Literal
77	ExprTuple	80
78	Lambda	parameter: 99 body: 81
79	Lambda	parameter: 99 body: 82
80	Lambda	parameter: 115 body: 83
81	Operation	operator: 93 operands: 84
82	Operation	operator: 109 operand: 90
83	Conditional	value: 86 condition: 87
84	ExprTuple	88, 89
85	ExprTuple	90
86	Operation	operator: 91 operands: 92
87	Operation	operator: 93 operands: 94
88	IndexedVar	variable: 100 index: 99
89	Literal
90	IndexedVar	variable: 113 index: 99
91	Literal
92	ExprTuple	96, 97
93	Literal
94	ExprTuple	115, 98
95	ExprTuple	99
96	IndexedVar	variable: 100 index: 115
97	Operation	operator: 101 operands: 102
98	Operation	operator: 103 operands: 104
99	Variable
100	Variable
101	Literal
102	ExprTuple	105, 106
103	Literal
104	ExprTuple	107, 108
105	Operation	operator: 109 operand: 112
106	Operation	operator: 110 operand: 112
107	Literal
108	Variable
109	Literal
110	Literal
111	ExprTuple	112
112	IndexedVar	variable: 113 index: 115
113	Variable
114	ExprTuple	115
115	Variable