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 Function, 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, VecSum
from proveit.logic import Equals, Forall, Implies, 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([a_1_to_i, c_1_to_k], 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))

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_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{k}\right) \mapsto \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)

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