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 K, Lambda, Q, V, f, 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([K, f, Q], Forall(instance_param_or_params = [j], instance_expr = Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [v], instance_expr = Implies(InSet(vec_summation_b1toj_fQ, K), Equals(VecSum(index_or_indices = sub_expr1, summand = ScalarMult(f__b_1_to_j, v), condition = Q__b_1_to_j), ScalarMult(Sum(index_or_indices = sub_expr1, summand = f__b_1_to_j, condition = Q__b_1_to_j), v)).with_wrapping_at(1)).with_wrapping_at(2), domain = V), 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\right) \mapsto \left[\forall_{j \in \mathbb{N}^+}~\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[\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] \\  = \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) \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	40, 59, 60
2	Operation	operator: 17 operand: 4
3	ExprTuple	4
4	Lambda	parameter: 65 body: 6
5	ExprTuple	65
6	Conditional	value: 7 condition: 8
7	Operation	operator: 17 operand: 11
8	Operation	operator: 35 operands: 10
9	ExprTuple	11
10	ExprTuple	65, 12
11	Lambda	parameter: 34 body: 14
12	Literal
13	ExprTuple	34
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	34, 22
21	Lambda	parameter: 56 body: 24
22	Operation	operator: 25 operand: 40
23	ExprTuple	56
24	Conditional	value: 27 condition: 28
25	Literal
26	ExprTuple	40
27	Operation	operator: 29 operands: 30
28	Operation	operator: 35 operands: 31
29	Literal
30	ExprTuple	32, 33
31	ExprTuple	56, 34
32	Operation	operator: 35 operands: 36
33	Operation	operator: 37 operands: 38
34	Variable
35	Literal
36	ExprTuple	39, 40
37	Literal
38	ExprTuple	41, 42
39	Operation	operator: 43 operand: 52
40	Variable
41	Operation	operator: 43 operand: 46
42	Operation	operator: 53 operands: 45
43	Literal
44	ExprTuple	46
45	ExprTuple	47, 56
46	Lambda	parameters: 61 body: 48
47	Operation	operator: 49 operand: 52
48	Conditional	value: 51 condition: 58
49	Literal
50	ExprTuple	52
51	Operation	operator: 53 operands: 54
52	Lambda	parameters: 61 body: 55
53	Literal
54	ExprTuple	57, 56
55	Conditional	value: 57 condition: 58
56	Variable
57	Operation	operator: 59 operands: 61
58	Operation	operator: 60 operands: 61
59	Variable
60	Variable
61	ExprTuple	62
62	ExprRange	lambda_map: 63 start_index: 64 end_index: 65
63	Lambda	parameter: 69 body: 66
64	Literal
65	Variable
66	IndexedVar	variable: 67 index: 69
67	Variable
68	ExprTuple	69
69	Variable