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, 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.linear_algebra.addition import vec_summation_b1toj_fQ
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import NaturalPos

# build up the expression from sub-expressions
expr = Lambda(j, Conditional(Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [k], instance_expr = Implies(InSet(vec_summation_b1toj_fQ, V), Equals(VecSum(index_or_indices = [b_1_to_j], summand = ScalarMult(k, f__b_1_to_j), condition = Q__b_1_to_j), ScalarMult(k, vec_summation_b1toj_fQ)).with_wrapping_at(1)).with_wrapping_at(2), domain = K), 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_{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)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\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(k \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)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\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: 59 body: 2
1	ExprTuple	59
2	Conditional	value: 3 condition: 4
3	Operation	operator: 13 operand: 7
4	Operation	operator: 31 operands: 6
5	ExprTuple	7
6	ExprTuple	59, 8
7	Lambda	parameter: 35 body: 10
8	Literal
9	ExprTuple	35
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	35, 18
17	Lambda	parameter: 50 body: 20
18	Operation	operator: 21 operand: 30
19	ExprTuple	50
20	Conditional	value: 23 condition: 24
21	Literal
22	ExprTuple	30
23	Operation	operator: 25 operands: 26
24	Operation	operator: 31 operands: 27
25	Literal
26	ExprTuple	28, 29
27	ExprTuple	50, 30
28	Operation	operator: 31 operands: 32
29	Operation	operator: 33 operands: 34
30	Variable
31	Literal
32	ExprTuple	41, 35
33	Literal
34	ExprTuple	36, 37
35	Variable
36	Operation	operator: 43 operand: 40
37	Operation	operator: 47 operands: 39
38	ExprTuple	40
39	ExprTuple	50, 41
40	Lambda	parameters: 55 body: 42
41	Operation	operator: 43 operand: 46
42	Conditional	value: 45 condition: 52
43	Literal
44	ExprTuple	46
45	Operation	operator: 47 operands: 48
46	Lambda	parameters: 55 body: 49
47	Literal
48	ExprTuple	50, 51
49	Conditional	value: 51 condition: 52
50	Variable
51	Operation	operator: 53 operands: 55
52	Operation	operator: 54 operands: 55
53	Variable
54	Variable
55	ExprTuple	56
56	ExprRange	lambda_map: 57 start_index: 58 end_index: 59
57	Lambda	parameter: 63 body: 60
58	Literal
59	Variable
60	IndexedVar	variable: 61 index: 63
61	Variable
62	ExprTuple	63
63	Variable