Expression of type Conditional

from the theory of proveit.linear_algebra.inner_products

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, ExprRange, IndexedVar, Variable, i, lambda_, m, n, v
from proveit.core_expr_types import a_1_to_m, a_i, b_1_to_n, b_i, lambda_i
from proveit.linear_algebra import Dim, HilbertSpaces, OrthoNormBases, ScalarMult, TensorProd, VecSum
from proveit.logic import And, Equals, Exists, Forall, InClass, InSet, Set
from proveit.numbers import Interval, Min, RealNonNeg, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Min(m, n)
expr = Conditional(Forall(instance_param_or_params = [v], instance_expr = Exists(instance_param_or_params = [a_1_to_m], instance_expr = Exists(instance_param_or_params = [b_1_to_n], instance_expr = Exists(instance_param_or_params = [ExprRange(sub_expr1, IndexedVar(lambda_, sub_expr1), one, sub_expr2)], instance_expr = Equals(v, VecSum(index_or_indices = [i], summand = ScalarMult(lambda_i, TensorProd(a_i, b_i)), domain = Interval(one, sub_expr2))), domain = RealNonNeg).with_wrapping(), condition = InSet(Set(b_1_to_n), OrthoNormBases(B))).with_wrapping(), condition = InSet(Set(a_1_to_m), OrthoNormBases(A))).with_wrapping(), domain = TensorProd(A, B)), And(InClass(A, HilbertSpaces), InClass(B, HilbertSpaces), Equals(Dim(A), m), Equals(Dim(B), 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())

\left\{\forall_{v \in A {\otimes} B}~\left[\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{m}~|~\left\{a_{1}, a_{2}, \ldots, a_{m}\right\} \in \textrm{O.N.Bases}\left(A\right)}~\\
\left[\begin{array}{l}\exists_{b_{1}, b_{2}, \ldots, b_{n}~|~\left\{b_{1}, b_{2}, \ldots, b_{n}\right\} \in \textrm{O.N.Bases}\left(B\right)}~\\
\left[\begin{array}{l}\exists_{\lambda_{1}, \lambda_{2}, \ldots, \lambda_{{\rm Min}\left(m, n\right)} \in \mathbb{R}^{\ge 0}}~\\
\left(v = \left(\sum_{i=1}^{{\rm Min}\left(m, n\right)} \left(\lambda_{i} \cdot \left(a_{i} {\otimes} b_{i}\right)\right)\right)\right)\end{array}\right]\end{array}\right]\end{array}\right] \textrm{ if } A \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces} ,  B \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces} ,  \textrm{Dim}\left(A\right) = m ,  \textrm{Dim}\left(B\right) = n\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 6
2	Operation	operator: 66 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	7, 8, 9, 10
6	Lambda	parameter: 69 body: 12
7	Operation	operator: 14 operands: 13
8	Operation	operator: 14 operands: 15
9	Operation	operator: 64 operands: 16
10	Operation	operator: 64 operands: 17
11	ExprTuple	69
12	Conditional	value: 18 condition: 19
13	ExprTuple	43, 20
14	Literal
15	ExprTuple	62, 20
16	ExprTuple	21, 108
17	ExprTuple	22, 109
18	Operation	operator: 44 operand: 26
19	Operation	operator: 86 operands: 24
20	Literal
21	Operation	operator: 25 operand: 43
22	Operation	operator: 25 operand: 62
23	ExprTuple	26
24	ExprTuple	69, 27
25	Literal
26	Lambda	parameters: 38 body: 28
27	Operation	operator: 94 operands: 29
28	Conditional	value: 30 condition: 31
29	ExprTuple	43, 62
30	Operation	operator: 44 operand: 34
31	Operation	operator: 86 operands: 33
32	ExprTuple	34
33	ExprTuple	35, 36
34	Lambda	parameters: 55 body: 37
35	Operation	operator: 54 operands: 38
36	Operation	operator: 56 operand: 43
37	Conditional	value: 40 condition: 41
38	ExprTuple	42
39	ExprTuple	43
40	Operation	operator: 44 operand: 48
41	Operation	operator: 86 operands: 46
42	ExprRange	lambda_map: 47 start_index: 100 end_index: 108
43	Variable
44	Literal
45	ExprTuple	48
46	ExprTuple	49, 50
47	Lambda	parameter: 92 body: 51
48	Lambda	parameters: 52 body: 53
49	Operation	operator: 54 operands: 55
50	Operation	operator: 56 operand: 62
51	IndexedVar	variable: 102 index: 92
52	ExprTuple	58
53	Conditional	value: 59 condition: 60
54	Literal
55	ExprTuple	61
56	Literal
57	ExprTuple	62
58	ExprRange	lambda_map: 63 start_index: 100 end_index: 101
59	Operation	operator: 64 operands: 65
60	Operation	operator: 66 operands: 67
61	ExprRange	lambda_map: 68 start_index: 100 end_index: 109
62	Variable
63	Lambda	parameter: 92 body: 82
64	Literal
65	ExprTuple	69, 70
66	Literal
67	ExprTuple	71
68	Lambda	parameter: 92 body: 72
69	Variable
70	Operation	operator: 73 operand: 76
71	ExprRange	lambda_map: 75 start_index: 100 end_index: 101
72	IndexedVar	variable: 103 index: 92
73	Literal
74	ExprTuple	76
75	Lambda	parameter: 92 body: 77
76	Lambda	parameter: 107 body: 78
77	Operation	operator: 86 operands: 79
78	Conditional	value: 80 condition: 81
79	ExprTuple	82, 83
80	Operation	operator: 84 operands: 85
81	Operation	operator: 86 operands: 87
82	IndexedVar	variable: 93 index: 92
83	Literal
84	Literal
85	ExprTuple	89, 90
86	Literal
87	ExprTuple	107, 91
88	ExprTuple	92
89	IndexedVar	variable: 93 index: 107
90	Operation	operator: 94 operands: 95
91	Operation	operator: 96 operands: 97
92	Variable
93	Variable
94	Literal
95	ExprTuple	98, 99
96	Literal
97	ExprTuple	100, 101
98	IndexedVar	variable: 102 index: 107
99	IndexedVar	variable: 103 index: 107
100	Literal
101	Operation	operator: 105 operands: 106
102	Variable
103	Variable
104	ExprTuple	107
105	Literal
106	ExprTuple	108, 109
107	Variable
108	Variable
109	Variable