Expression of type Lambda

from the theory of proveit.linear_algebra.scalar_multiplication

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, j, v
from proveit.core_expr_types import Q__b_1_to_j, b_1_to_j, f__b_1_to_j
from proveit.linear_algebra import ScalarMult, VecSpaces, VecSum
from proveit.linear_algebra.addition import vec_summation_b1toj_fQ
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import NaturalPos, Sum

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
expr = Lambda(j, Conditional(Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [v], instance_expr = Implies(InSet(vec_summation_b1toj_fQ, K), Equals(ScalarMult(Sum(index_or_indices = sub_expr1, summand = f__b_1_to_j, condition = Q__b_1_to_j), v), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(f__b_1_to_j, v), condition = Q__b_1_to_j)).with_wrapping_at(1)).with_wrapping_at(2), domain = V), domain = VecSpaces(K)), InSet(j, NaturalPos)))

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())

j \mapsto \left\{\forall_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{v \in V}~\left(\begin{array}{c} \begin{array}{l} \left(\left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right] \in K\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(\left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right] \cdot v\right) \\  = \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(f\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot v\right)\right] \end{array} \end{array}\right) \end{array} \end{array}\right)\right] \textrm{ if } j \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	parameter: 61 body: 2
1	ExprTuple	61
2	Conditional	value: 3 condition: 4
3	Operation	operator: 13 operand: 7
4	Operation	operator: 31 operands: 6
5	ExprTuple	7
6	ExprTuple	61, 8
7	Lambda	parameter: 30 body: 10
8	Literal
9	ExprTuple	30
10	Conditional	value: 11 condition: 12
11	Operation	operator: 13 operand: 17
12	Operation	operator: 15 operands: 16
13	Literal
14	ExprTuple	17
15	Literal
16	ExprTuple	30, 18
17	Lambda	parameter: 54 body: 20
18	Operation	operator: 21 operand: 36
19	ExprTuple	54
20	Conditional	value: 23 condition: 24
21	Literal
22	ExprTuple	36
23	Operation	operator: 25 operands: 26
24	Operation	operator: 31 operands: 27
25	Literal
26	ExprTuple	28, 29
27	ExprTuple	54, 30
28	Operation	operator: 31 operands: 32
29	Operation	operator: 33 operands: 34
30	Variable
31	Literal
32	ExprTuple	35, 36
33	Literal
34	ExprTuple	37, 38
35	Operation	operator: 40 operand: 47
36	Variable
37	Operation	operator: 50 operands: 39
38	Operation	operator: 40 operand: 43
39	ExprTuple	42, 54
40	Literal
41	ExprTuple	43
42	Operation	operator: 44 operand: 47
43	Lambda	parameters: 57 body: 46
44	Literal
45	ExprTuple	47
46	Conditional	value: 48 condition: 52
47	Lambda	parameters: 57 body: 49
48	Operation	operator: 50 operands: 51
49	Conditional	value: 53 condition: 52
50	Literal
51	ExprTuple	53, 54
52	Operation	operator: 55 operands: 57
53	Operation	operator: 56 operands: 57
54	Variable
55	Variable
56	Variable
57	ExprTuple	58
58	ExprRange	lambda_map: 59 start_index: 60 end_index: 61
59	Lambda	parameter: 65 body: 62
60	Literal
61	Variable
62	IndexedVar	variable: 63 index: 65
63	Variable
64	ExprTuple	65
65	Variable