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, ExprRange, IndexedVar, K, Lambda, U, V, Variable, W, 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, InClass
from proveit.numbers import one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = VecSpaces(K)
sub_expr3 = [U_1_to_i, V, W_1_to_k]
expr = Lambda(sub_expr3, Conditional(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_expr3).with_wrapping(), And(ExprRange(sub_expr1, InClass(IndexedVar(U, sub_expr1), sub_expr2), one, i), InClass(V, sub_expr2), ExprRange(sub_expr1, InClass(IndexedVar(W, sub_expr1), sub_expr2), one, 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())

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