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, Function, K, Lambda, V, s
from proveit.core_expr_types import Q__b_1_to_j, a_1_to_i, b_1_to_j, c_1_to_k, f__b_1_to_j
from proveit.linear_algebra import ScalarMult, TensorProd, VecSpaces, VecSum
from proveit.logic import Equals, Forall, Implies, InClass, InSet

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
sub_expr2 = Function(s, sub_expr1)
sub_expr3 = ScalarMult(sub_expr2, f__b_1_to_j)
expr = Lambda(V, Conditional(Forall(instance_param_or_params = [a_1_to_i, c_1_to_k], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(TensorProd(a_1_to_i, sub_expr3, c_1_to_k), V), condition = Q__b_1_to_j), Equals(VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr2, TensorProd(a_1_to_i, f__b_1_to_j, c_1_to_k)), condition = Q__b_1_to_j), TensorProd(a_1_to_i, VecSum(index_or_indices = sub_expr1, summand = sub_expr3, condition = Q__b_1_to_j), c_1_to_k)).with_wrapping_at(1)).with_wrapping_at(2)).with_wrapping(), InClass(V, VecSpaces(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())

V \mapsto \left\{\begin{array}{l}\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{k}}~\\
\left(\begin{array}{c} \begin{array}{l} \left[\forall_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \in V\right)\right] \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} f\left(b_{1}, b_{2}, \ldots, b_{j}\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right)\right)\right] \\  = \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(s\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right)\right]{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \end{array} \end{array}\right) \end{array} \end{array}\right)\end{array} \textrm{ if } V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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