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 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(alpha, Conditional(Forall(instance_param_or_params = [i, k], instance_expr = 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(), domain = Natural), InSet(alpha, K)))

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())

\alpha \mapsto \left\{\forall_{i, k \in \mathbb{N}}~\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}\right] \textrm{ if } \alpha \in K\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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