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 Function, K, Lambda, Q, V, c, f, j, 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, InSet
from proveit.numbers import Mult, NaturalPos

# 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([K, f, Q, c], Forall(instance_param_or_params = [j], instance_expr = Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [k], instance_expr = Implies(InSet(sub_expr3, V), Equals(VecSum(index_or_indices = sub_expr1, summand = ScalarMult(Mult(k, sub_expr2), f__b_1_to_j), condition = Q__b_1_to_j), ScalarMult(k, sub_expr3)).with_wrapping_at(1)).with_wrapping_at(1), domain = K), domain = VecSpaces(K)), domain = 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())

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