Expression of type Lambda

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, 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 = 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(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..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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