Expression of type Iff

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, 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 Commutator, Hspace, LinMap, OrthoNormBases, ScalarMult, VecSum, VecZero
from proveit.logic import And, Equals, Exists, Iff, 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 = Iff(Equals(Commutator(A, B), VecZero(LinMap(Hspace, Hspace))), 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_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(), condition = InSet(Set(v_1_to_n), OrthoNormBases(Hspace))).with_wrapping()).with_wrapping_at(2)

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())

\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}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(2)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

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