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, InSet, Set
from proveit.numbers import Interval, Min, NaturalPos, 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 = [A, B], instance_expr = 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)), domain = HilbertSpaces, conditions = [Equals(Dim(A), m), Equals(Dim(B), n)]).with_wrapping(), And(InSet(m, NaturalPos), InSet(n, NaturalPos)))

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\{\begin{array}{l}\forall_{A, B \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}~|~\textrm{Dim}\left(A\right) = m, \textrm{Dim}\left(B\right) = n}~\\
\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]\right]\end{array} \textrm{ if } m \in \mathbb{N}^+ ,  n \in \mathbb{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: 14 operand: 5
2	Operation	operator: 77 operands: 4
3	ExprTuple	5
4	ExprTuple	6, 7
5	Lambda	parameters: 40 body: 8
6	Operation	operator: 97 operands: 9
7	Operation	operator: 97 operands: 10
8	Conditional	value: 11 condition: 12
9	ExprTuple	119, 13
10	ExprTuple	120, 13
11	Operation	operator: 14 operand: 17
12	Operation	operator: 77 operands: 16
13	Literal
14	Literal
15	ExprTuple	17
16	ExprTuple	18, 19, 20, 21
17	Lambda	parameter: 80 body: 23
18	Operation	operator: 25 operands: 24
19	Operation	operator: 25 operands: 26
20	Operation	operator: 75 operands: 27
21	Operation	operator: 75 operands: 28
22	ExprTuple	80
23	Conditional	value: 29 condition: 30
24	ExprTuple	54, 31
25	Literal
26	ExprTuple	73, 31
27	ExprTuple	32, 119
28	ExprTuple	33, 120
29	Operation	operator: 55 operand: 37
30	Operation	operator: 97 operands: 35
31	Literal
32	Operation	operator: 36 operand: 54
33	Operation	operator: 36 operand: 73
34	ExprTuple	37
35	ExprTuple	80, 38
36	Literal
37	Lambda	parameters: 49 body: 39
38	Operation	operator: 105 operands: 40
39	Conditional	value: 41 condition: 42
40	ExprTuple	54, 73
41	Operation	operator: 55 operand: 45
42	Operation	operator: 97 operands: 44
43	ExprTuple	45
44	ExprTuple	46, 47
45	Lambda	parameters: 66 body: 48
46	Operation	operator: 65 operands: 49
47	Operation	operator: 67 operand: 54
48	Conditional	value: 51 condition: 52
49	ExprTuple	53
50	ExprTuple	54
51	Operation	operator: 55 operand: 59
52	Operation	operator: 97 operands: 57
53	ExprRange	lambda_map: 58 start_index: 111 end_index: 119
54	Variable
55	Literal
56	ExprTuple	59
57	ExprTuple	60, 61
58	Lambda	parameter: 103 body: 62
59	Lambda	parameters: 63 body: 64
60	Operation	operator: 65 operands: 66
61	Operation	operator: 67 operand: 73
62	IndexedVar	variable: 113 index: 103
63	ExprTuple	69
64	Conditional	value: 70 condition: 71
65	Literal
66	ExprTuple	72
67	Literal
68	ExprTuple	73
69	ExprRange	lambda_map: 74 start_index: 111 end_index: 112
70	Operation	operator: 75 operands: 76
71	Operation	operator: 77 operands: 78
72	ExprRange	lambda_map: 79 start_index: 111 end_index: 120
73	Variable
74	Lambda	parameter: 103 body: 93
75	Literal
76	ExprTuple	80, 81
77	Literal
78	ExprTuple	82
79	Lambda	parameter: 103 body: 83
80	Variable
81	Operation	operator: 84 operand: 87
82	ExprRange	lambda_map: 86 start_index: 111 end_index: 112
83	IndexedVar	variable: 114 index: 103
84	Literal
85	ExprTuple	87
86	Lambda	parameter: 103 body: 88
87	Lambda	parameter: 118 body: 89
88	Operation	operator: 97 operands: 90
89	Conditional	value: 91 condition: 92
90	ExprTuple	93, 94
91	Operation	operator: 95 operands: 96
92	Operation	operator: 97 operands: 98
93	IndexedVar	variable: 104 index: 103
94	Literal
95	Literal
96	ExprTuple	100, 101
97	Literal
98	ExprTuple	118, 102
99	ExprTuple	103
100	IndexedVar	variable: 104 index: 118
101	Operation	operator: 105 operands: 106
102	Operation	operator: 107 operands: 108
103	Variable
104	Variable
105	Literal
106	ExprTuple	109, 110
107	Literal
108	ExprTuple	111, 112
109	IndexedVar	variable: 113 index: 118
110	IndexedVar	variable: 114 index: 118
111	Literal
112	Operation	operator: 116 operands: 117
113	Variable
114	Variable
115	ExprTuple	118
116	Literal
117	ExprTuple	119, 120
118	Variable
119	Variable
120	Variable