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 A, B, i, n
from proveit.core_expr_types import a_1_to_n, a_i, b_1_to_n, b_i, v_i
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import And, Equals, Exists
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 = 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()

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_{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}

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