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