Expression of type Lambda

from the theory of proveit.linear_algebra.tensors

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 Conditional, K, Lambda, V, alpha, b, i, k
from proveit.core_expr_types import U_1_to_i, W_1_to_k, a_1_to_i, c_1_to_k
from proveit.linear_algebra import ScalarMult, TensorProd, VecSpaces
from proveit.logic import And, Equals, Forall, InSet
from proveit.numbers import Natural

# build up the expression from sub-expressions
sub_expr1 = [U_1_to_i, V, W_1_to_k]
expr = Lambda([i, k], Conditional(Forall(instance_param_or_params = sub_expr1, instance_expr = Forall(instance_param_or_params = [a_1_to_i, b, c_1_to_k], instance_expr = Equals(TensorProd(a_1_to_i, ScalarMult(alpha, b), c_1_to_k), ScalarMult(alpha, TensorProd(a_1_to_i, b, c_1_to_k))).with_wrapping_at(1), domains = sub_expr1).with_wrapping(), domain = VecSpaces(K)).with_wrapping(), And(InSet(i, Natural), InSet(k, Natural))))

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(i, k\right) \mapsto \left\{\begin{array}{l}\forall_{U_{1}, U_{2}, \ldots, U_{i}, V, W_{1}, W_{2}, \ldots, W_{k} \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\\
\left[\begin{array}{l}\forall_{\left(a_{1} \in U_{1}\right), \left(a_{2} \in U_{2}\right), \ldots, \left(a_{i} \in U_{i}\right), b \in V,\left(c_{1} \in W_{1}\right), \left(c_{2} \in W_{2}\right), \ldots, \left(c_{k} \in W_{k}\right)}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left(\alpha \cdot b\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \\  = \left(\alpha \cdot \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right)\right) \end{array} \end{array}\right)\end{array}\right]\end{array} \textrm{ if } i \in \mathbb{N} ,  k \in \mathbb{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: 1 body: 2
1	ExprTuple	76, 79
2	Conditional	value: 3 condition: 4
3	Operation	operator: 21 operand: 7
4	Operation	operator: 38 operands: 6
5	ExprTuple	7
6	ExprTuple	8, 9
7	Lambda	parameters: 10 body: 11
8	Operation	operator: 67 operands: 12
9	Operation	operator: 67 operands: 13
10	ExprTuple	14, 59, 15
11	Conditional	value: 16 condition: 17
12	ExprTuple	76, 18
13	ExprTuple	79, 18
14	ExprRange	lambda_map: 19 start_index: 78 end_index: 76
15	ExprRange	lambda_map: 20 start_index: 78 end_index: 79
16	Operation	operator: 21 operand: 24
17	Operation	operator: 38 operands: 23
18	Literal
19	Lambda	parameter: 87 body: 73
20	Lambda	parameter: 87 body: 74
21	Literal
22	ExprTuple	24
23	ExprTuple	25, 26, 27
24	Lambda	parameters: 65 body: 28
25	ExprRange	lambda_map: 29 start_index: 78 end_index: 76
26	Operation	operator: 41 operands: 30
27	ExprRange	lambda_map: 31 start_index: 78 end_index: 79
28	Conditional	value: 32 condition: 33
29	Lambda	parameter: 87 body: 34
30	ExprTuple	59, 48
31	Lambda	parameter: 87 body: 35
32	Operation	operator: 36 operands: 37
33	Operation	operator: 38 operands: 39
34	Operation	operator: 41 operands: 40
35	Operation	operator: 41 operands: 42
36	Literal
37	ExprTuple	43, 44
38	Literal
39	ExprTuple	45, 46, 47
40	ExprTuple	73, 48
41	Literal
42	ExprTuple	74, 48
43	Operation	operator: 64 operands: 49
44	Operation	operator: 62 operands: 50
45	ExprRange	lambda_map: 51 start_index: 78 end_index: 76
46	Operation	operator: 67 operands: 52
47	ExprRange	lambda_map: 53 start_index: 78 end_index: 79
48	Operation	operator: 54 operand: 61
49	ExprTuple	70, 56, 72
50	ExprTuple	69, 57
51	Lambda	parameter: 87 body: 58
52	ExprTuple	71, 59
53	Lambda	parameter: 87 body: 60
54	Literal
55	ExprTuple	61
56	Operation	operator: 62 operands: 63
57	Operation	operator: 64 operands: 65
58	Operation	operator: 67 operands: 66
59	Variable
60	Operation	operator: 67 operands: 68
61	Variable
62	Literal
63	ExprTuple	69, 71
64	Literal
65	ExprTuple	70, 71, 72
66	ExprTuple	82, 73
67	Literal
68	ExprTuple	83, 74
69	Variable
70	ExprRange	lambda_map: 75 start_index: 78 end_index: 76
71	Variable
72	ExprRange	lambda_map: 77 start_index: 78 end_index: 79
73	IndexedVar	variable: 80 index: 87
74	IndexedVar	variable: 81 index: 87
75	Lambda	parameter: 87 body: 82
76	Variable
77	Lambda	parameter: 87 body: 83
78	Literal
79	Variable
80	Variable
81	Variable
82	IndexedVar	variable: 84 index: 87
83	IndexedVar	variable: 85 index: 87
84	Variable
85	Variable
86	ExprTuple	87
87	Variable