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 ExprRange, IndexedVar, Variable, i, lambda_, m, n
from proveit.core_expr_types import a_i, b_i, lambda_i
from proveit.linear_algebra import ScalarMult, VecSum
from proveit.logic import Equals, Exists
from proveit.numbers import Interval, Min, RealNonNeg, one
from proveit.physics.quantum import Ket, Qmult, var_ket_psi

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Min(m, n)
expr = Exists(instance_param_or_params = [ExprRange(sub_expr1, IndexedVar(lambda_, sub_expr1), one, sub_expr2)], instance_expr = Equals(var_ket_psi, VecSum(index_or_indices = [i], summand = ScalarMult(lambda_i, Qmult(Ket(a_i), Ket(b_i))), domain = Interval(one, sub_expr2))), domain = RealNonNeg).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_{\lambda_{1}, \lambda_{2}, \ldots, \lambda_{{\rm Min}\left(m, n\right)} \in \mathbb{R}^{\ge 0}}~\\
\left(\lvert \psi \rangle = \left(\sum_{i=1}^{{\rm Min}\left(m, n\right)} \left(\lambda_{i} \cdot \left(\lvert a_{i} \rangle \thinspace \lvert b_{i} \rangle\right)\right)\right)\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
5	Conditional	value: 7 condition: 8
6	ExprRange	lambda_map: 9 start_index: 44 end_index: 45
7	Operation	operator: 10 operands: 11
8	Operation	operator: 12 operands: 13
9	Lambda	parameter: 36 body: 26
10	Literal
11	ExprTuple	14, 15
12	Literal
13	ExprTuple	16
14	Variable
15	Operation	operator: 17 operand: 20
16	ExprRange	lambda_map: 19 start_index: 44 end_index: 45
17	Literal
18	ExprTuple	20
19	Lambda	parameter: 36 body: 21
20	Lambda	parameter: 58 body: 22
21	Operation	operator: 30 operands: 23
22	Conditional	value: 24 condition: 25
23	ExprTuple	26, 27
24	Operation	operator: 28 operands: 29
25	Operation	operator: 30 operands: 31
26	IndexedVar	variable: 37 index: 36
27	Literal
28	Literal
29	ExprTuple	33, 34
30	Literal
31	ExprTuple	58, 35
32	ExprTuple	36
33	IndexedVar	variable: 37 index: 58
34	Operation	operator: 38 operands: 39
35	Operation	operator: 40 operands: 41
36	Variable
37	Variable
38	Literal
39	ExprTuple	42, 43
40	Literal
41	ExprTuple	44, 45
42	Operation	operator: 47 operand: 51
43	Operation	operator: 47 operand: 52
44	Literal
45	Operation	operator: 49 operands: 50
46	ExprTuple	51
47	Literal
48	ExprTuple	52
49	Literal
50	ExprTuple	53, 54
51	IndexedVar	variable: 55 index: 58
52	IndexedVar	variable: 56 index: 58
53	Variable
54	Variable
55	Variable
56	Variable
57	ExprTuple	58
58	Variable