Expression of type ExprTuple

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, ExprTuple, IndexedVar, Lambda, 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 = ExprTuple(Lambda([m, n], 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(\left(m, n\right) \mapsto \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..\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	ExprTuple	1
1	Lambda	parameters: 119 body: 2
2	Conditional	value: 3 condition: 4
3	Operation	operator: 16 operand: 7
4	Operation	operator: 79 operands: 6
5	ExprTuple	7
6	ExprTuple	8, 9
7	Lambda	parameters: 42 body: 10
8	Operation	operator: 99 operands: 11
9	Operation	operator: 99 operands: 12
10	Conditional	value: 13 condition: 14
11	ExprTuple	121, 15
12	ExprTuple	122, 15
13	Operation	operator: 16 operand: 19
14	Operation	operator: 79 operands: 18
15	Literal
16	Literal
17	ExprTuple	19
18	ExprTuple	20, 21, 22, 23
19	Lambda	parameter: 82 body: 25
20	Operation	operator: 27 operands: 26
21	Operation	operator: 27 operands: 28
22	Operation	operator: 77 operands: 29
23	Operation	operator: 77 operands: 30
24	ExprTuple	82
25	Conditional	value: 31 condition: 32
26	ExprTuple	56, 33
27	Literal
28	ExprTuple	75, 33
29	ExprTuple	34, 121
30	ExprTuple	35, 122
31	Operation	operator: 57 operand: 39
32	Operation	operator: 99 operands: 37
33	Literal
34	Operation	operator: 38 operand: 56
35	Operation	operator: 38 operand: 75
36	ExprTuple	39
37	ExprTuple	82, 40
38	Literal
39	Lambda	parameters: 51 body: 41
40	Operation	operator: 107 operands: 42
41	Conditional	value: 43 condition: 44
42	ExprTuple	56, 75
43	Operation	operator: 57 operand: 47
44	Operation	operator: 99 operands: 46
45	ExprTuple	47
46	ExprTuple	48, 49
47	Lambda	parameters: 68 body: 50
48	Operation	operator: 67 operands: 51
49	Operation	operator: 69 operand: 56
50	Conditional	value: 53 condition: 54
51	ExprTuple	55
52	ExprTuple	56
53	Operation	operator: 57 operand: 61
54	Operation	operator: 99 operands: 59
55	ExprRange	lambda_map: 60 start_index: 113 end_index: 121
56	Variable
57	Literal
58	ExprTuple	61
59	ExprTuple	62, 63
60	Lambda	parameter: 105 body: 64
61	Lambda	parameters: 65 body: 66
62	Operation	operator: 67 operands: 68
63	Operation	operator: 69 operand: 75
64	IndexedVar	variable: 115 index: 105
65	ExprTuple	71
66	Conditional	value: 72 condition: 73
67	Literal
68	ExprTuple	74
69	Literal
70	ExprTuple	75
71	ExprRange	lambda_map: 76 start_index: 113 end_index: 114
72	Operation	operator: 77 operands: 78
73	Operation	operator: 79 operands: 80
74	ExprRange	lambda_map: 81 start_index: 113 end_index: 122
75	Variable
76	Lambda	parameter: 105 body: 95
77	Literal
78	ExprTuple	82, 83
79	Literal
80	ExprTuple	84
81	Lambda	parameter: 105 body: 85
82	Variable
83	Operation	operator: 86 operand: 89
84	ExprRange	lambda_map: 88 start_index: 113 end_index: 114
85	IndexedVar	variable: 116 index: 105
86	Literal
87	ExprTuple	89
88	Lambda	parameter: 105 body: 90
89	Lambda	parameter: 120 body: 91
90	Operation	operator: 99 operands: 92
91	Conditional	value: 93 condition: 94
92	ExprTuple	95, 96
93	Operation	operator: 97 operands: 98
94	Operation	operator: 99 operands: 100
95	IndexedVar	variable: 106 index: 105
96	Literal
97	Literal
98	ExprTuple	102, 103
99	Literal
100	ExprTuple	120, 104
101	ExprTuple	105
102	IndexedVar	variable: 106 index: 120
103	Operation	operator: 107 operands: 108
104	Operation	operator: 109 operands: 110
105	Variable
106	Variable
107	Literal
108	ExprTuple	111, 112
109	Literal
110	ExprTuple	113, 114
111	IndexedVar	variable: 115 index: 120
112	IndexedVar	variable: 116 index: 120
113	Literal
114	Operation	operator: 118 operands: 119
115	Variable
116	Variable
117	ExprTuple	120
118	Literal
119	ExprTuple	121, 122
120	Variable
121	Variable
122	Variable