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, Function, K, Lambda, V, c, k
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.logic import Equals, Forall, Implies, InClass, InSet
from proveit.numbers import Mult

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
sub_expr2 = Function(c, sub_expr1)
sub_expr3 = VecSum(index_or_indices = sub_expr1, summand = ScalarMult(sub_expr2, f__b_1_to_j), condition = Q__b_1_to_j)
expr = Lambda(V, Conditional(Forall(instance_param_or_params = [k], instance_expr = Implies(InSet(sub_expr3, V), Equals(ScalarMult(k, sub_expr3), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(Mult(k, sub_expr2), f__b_1_to_j), condition = Q__b_1_to_j)).with_wrapping_at(1)).with_wrapping_at(1), domain = K), 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\{\forall_{k \in K}~\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)}~\left(c\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right)\right] \in V\right) \\  \Rightarrow \left(\begin{array}{c} \begin{array}{l} \left(k \cdot \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(c\left(b_{1}, b_{2}, \ldots, b_{j}\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right)\right]\right) \\  = \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(\left(k \cdot c\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right) \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right)\right] \end{array} \end{array}\right) \end{array} \end{array}\right) \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: 27 body: 2
1	ExprTuple	27
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 9
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	27, 10
9	Lambda	parameter: 50 body: 12
10	Operation	operator: 13 operand: 22
11	ExprTuple	50
12	Conditional	value: 15 condition: 16
13	Literal
14	ExprTuple	22
15	Operation	operator: 17 operands: 18
16	Operation	operator: 23 operands: 19
17	Literal
18	ExprTuple	20, 21
19	ExprTuple	50, 22
20	Operation	operator: 23 operands: 24
21	Operation	operator: 25 operands: 26
22	Variable
23	Literal
24	ExprTuple	32, 27
25	Literal
26	ExprTuple	28, 29
27	Variable
28	Operation	operator: 44 operands: 30
29	Operation	operator: 34 operand: 33
30	ExprTuple	50, 32
31	ExprTuple	33
32	Operation	operator: 34 operand: 37
33	Lambda	parameters: 54 body: 36
34	Literal
35	ExprTuple	37
36	Conditional	value: 38 condition: 42
37	Lambda	parameters: 54 body: 39
38	Operation	operator: 44 operands: 40
39	Conditional	value: 41 condition: 42
40	ExprTuple	43, 49
41	Operation	operator: 44 operands: 45
42	Operation	operator: 46 operands: 54
43	Operation	operator: 47 operands: 48
44	Literal
45	ExprTuple	51, 49
46	Variable
47	Literal
48	ExprTuple	50, 51
49	Operation	operator: 52 operands: 54
50	Variable
51	Operation	operator: 53 operands: 54
52	Variable
53	Variable
54	ExprTuple	55
55	ExprRange	lambda_map: 56 start_index: 57 end_index: 58
56	Lambda	parameter: 62 body: 59
57	Literal
58	Variable
59	IndexedVar	variable: 60 index: 62
60	Variable
61	ExprTuple	62
62	Variable