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 Adj, Commutator, Dim, HilbertSpaces, Hspace, LinMap, OrthoNormBases, ScalarMult, VecSum, VecZero
from proveit.logic import And, Equals, Exists, Forall, Iff, InClass, 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 = LinMap(Hspace, Hspace)
sub_expr3 = Interval(one, n)
sub_expr4 = Qmult(Ket(v_i), Bra(v_i))
expr = Lambda(Hspace, Conditional(Forall(instance_param_or_params = [A, B], instance_expr = Iff(Equals(Commutator(A, B), VecZero(sub_expr2)), Exists(instance_param_or_params = [v_1_to_n], instance_expr = 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_expr4), domain = sub_expr3)), Equals(B, VecSum(index_or_indices = sub_expr1, summand = ScalarMult(b_i, sub_expr4), domain = sub_expr3))).with_wrapping_at(2), domain = Complex).with_wrapping(), condition = InSet(Set(v_1_to_n), OrthoNormBases(Hspace))).with_wrapping()).with_wrapping_at(2), domain = sub_expr2, conditions = [Equals(Adj(A), A), Equals(Adj(B), B)]).with_wrapping(), And(InClass(Hspace, HilbertSpaces), Equals(Dim(Hspace), n))))

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())

\mathcal{H} \mapsto \left\{\begin{array}{l}\forall_{A, B \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)~|~A^{\dagger} = A, B^{\dagger} = B}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\left[A, B\right] = \vec{0}\left(\mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)\right)\right) \Leftrightarrow  \\ \left[\begin{array}{l}\exists_{v_{1}, v_{2}, \ldots, v_{n}~|~\left\{v_{1}, v_{2}, \ldots, v_{n}\right\} \in \textrm{O.N.Bases}\left(\mathcal{H}\right)}~\\
\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}\right]\end{array}\right] \end{array} \end{array}\right)\end{array} \textrm{ if } \mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces} ,  \textrm{Dim}\left(\mathcal{H}\right) = 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: 69 body: 1
1	Conditional	value: 2 condition: 3
2	Operation	operator: 4 operand: 7
3	Operation	operator: 73 operands: 6
4	Literal
5	ExprTuple	7
6	ExprTuple	8, 9
7	Lambda	parameters: 40 body: 10
8	Operation	operator: 11 operands: 12
9	Operation	operator: 82 operands: 13
10	Conditional	value: 14 condition: 15
11	Literal
12	ExprTuple	69, 16
13	ExprTuple	17, 127
14	Operation	operator: 18 operands: 19
15	Operation	operator: 73 operands: 20
16	Literal
17	Operation	operator: 21 operand: 69
18	Literal
19	ExprTuple	22, 23
20	ExprTuple	24, 25, 26, 27
21	Literal
22	Operation	operator: 82 operands: 28
23	Operation	operator: 52 operand: 36
24	Operation	operator: 112 operands: 30
25	Operation	operator: 112 operands: 31
26	Operation	operator: 82 operands: 32
27	Operation	operator: 82 operands: 33
28	ExprTuple	34, 35
29	ExprTuple	36
30	ExprTuple	86, 47
31	ExprTuple	88, 47
32	ExprTuple	37, 86
33	ExprTuple	38, 88
34	Operation	operator: 39 operands: 40
35	Operation	operator: 41 operand: 47
36	Lambda	parameters: 61 body: 43
37	Operation	operator: 45 operand: 86
38	Operation	operator: 45 operand: 88
39	Literal
40	ExprTuple	86, 88
41	Literal
42	ExprTuple	47
43	Conditional	value: 48 condition: 49
44	ExprTuple	86
45	Literal
46	ExprTuple	88
47	Operation	operator: 50 operands: 51
48	Operation	operator: 52 operand: 55
49	Operation	operator: 112 operands: 54
50	Literal
51	ExprTuple	69, 69
52	Literal
53	ExprTuple	55
54	ExprTuple	56, 57
55	Lambda	parameters: 58 body: 59
56	Operation	operator: 60 operands: 61
57	Operation	operator: 62 operand: 69
58	ExprTuple	64, 65
59	Conditional	value: 66 condition: 67
60	Literal
61	ExprTuple	68
62	Literal
63	ExprTuple	69
64	ExprRange	lambda_map: 70 start_index: 126 end_index: 127
65	ExprRange	lambda_map: 71 start_index: 126 end_index: 127
66	Operation	operator: 73 operands: 72
67	Operation	operator: 73 operands: 74
68	ExprRange	lambda_map: 75 start_index: 126 end_index: 127
69	Variable
70	Lambda	parameter: 108 body: 99
71	Lambda	parameter: 108 body: 100
72	ExprTuple	76, 77
73	Literal
74	ExprTuple	78, 79
75	Lambda	parameter: 108 body: 80
76	Operation	operator: 82 operands: 81
77	Operation	operator: 82 operands: 83
78	ExprRange	lambda_map: 84 start_index: 126 end_index: 127
79	ExprRange	lambda_map: 85 start_index: 126 end_index: 127
80	IndexedVar	variable: 132 index: 108
81	ExprTuple	86, 87
82	Literal
83	ExprTuple	88, 89
84	Lambda	parameter: 108 body: 90
85	Lambda	parameter: 108 body: 91
86	Variable
87	Operation	operator: 93 operand: 97
88	Variable
89	Operation	operator: 93 operand: 98
90	Operation	operator: 112 operands: 95
91	Operation	operator: 112 operands: 96
92	ExprTuple	97
93	Literal
94	ExprTuple	98
95	ExprTuple	99, 101
96	ExprTuple	100, 101
97	Lambda	parameter: 134 body: 102
98	Lambda	parameter: 134 body: 103
99	IndexedVar	variable: 118 index: 108
100	IndexedVar	variable: 119 index: 108
101	Literal
102	Conditional	value: 105 condition: 107
103	Conditional	value: 106 condition: 107
104	ExprTuple	108
105	Operation	operator: 110 operands: 109
106	Operation	operator: 110 operands: 111
107	Operation	operator: 112 operands: 113
108	Variable
109	ExprTuple	114, 116
110	Literal
111	ExprTuple	115, 116
112	Literal
113	ExprTuple	134, 117
114	IndexedVar	variable: 118 index: 134
115	IndexedVar	variable: 119 index: 134
116	Operation	operator: 120 operands: 121
117	Operation	operator: 122 operands: 123
118	Variable
119	Variable
120	Literal
121	ExprTuple	124, 125
122	Literal
123	ExprTuple	126, 127
124	Operation	operator: 128 operand: 131
125	Operation	operator: 129 operand: 131
126	Literal
127	Variable
128	Literal
129	Literal
130	ExprTuple	131
131	IndexedVar	variable: 132 index: 134
132	Variable
133	ExprTuple	134
134	Variable