Expression of type Exists

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 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 Hspace, OrthoNormBases, ScalarMult, VecSum
from proveit.logic import Equals, Exists, 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
expr = 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()

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}{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}

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	True	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 9 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	5
4	Conditional	value: 6 condition: 7
5	ExprRange	lambda_map: 8 start_index: 67 end_index: 68
6	Operation	operator: 9 operand: 12
7	Operation	operator: 53 operands: 11
8	Lambda	parameter: 59 body: 50
9	Literal
10	ExprTuple	12
11	ExprTuple	13, 14
12	Lambda	parameters: 15 body: 16
13	Operation	operator: 30 operand: 23
14	Operation	operator: 18 operand: 24
15	ExprTuple	20
16	Conditional	value: 21 condition: 22
17	ExprTuple	23
18	Literal
19	ExprTuple	24
20	ExprRange	lambda_map: 25 start_index: 67 end_index: 68
21	Operation	operator: 26 operands: 27
22	Operation	operator: 28 operands: 29
23	Operation	operator: 30 operands: 31
24	Variable
25	Lambda	parameter: 59 body: 48
26	Literal
27	ExprTuple	32, 33
28	Literal
29	ExprTuple	34
30	Literal
31	ExprTuple	35
32	Variable
33	Operation	operator: 36 operand: 40
34	ExprRange	lambda_map: 38 start_index: 67 end_index: 68
35	ExprRange	lambda_map: 39 start_index: 67 end_index: 68
36	Literal
37	ExprTuple	40
38	Lambda	parameter: 59 body: 41
39	Lambda	parameter: 59 body: 42
40	Lambda	parameter: 75 body: 43
41	Operation	operator: 53 operands: 44
42	Operation	operator: 69 operand: 50
43	Conditional	value: 46 condition: 47
44	ExprTuple	48, 49
45	ExprTuple	50
46	Operation	operator: 51 operands: 52
47	Operation	operator: 53 operands: 54
48	IndexedVar	variable: 60 index: 59
49	Literal
50	IndexedVar	variable: 73 index: 59
51	Literal
52	ExprTuple	56, 57
53	Literal
54	ExprTuple	75, 58
55	ExprTuple	59
56	IndexedVar	variable: 60 index: 75
57	Operation	operator: 61 operands: 62
58	Operation	operator: 63 operands: 64
59	Variable
60	Variable
61	Literal
62	ExprTuple	65, 66
63	Literal
64	ExprTuple	67, 68
65	Operation	operator: 69 operand: 72
66	Operation	operator: 70 operand: 72
67	Literal
68	Variable
69	Literal
70	Literal
71	ExprTuple	72
72	IndexedVar	variable: 73 index: 75
73	Variable
74	ExprTuple	75
75	Variable