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, 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(j, Conditional(Forall(instance_param_or_params = [V], instance_expr = 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), 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)}~\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)\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: 66 body: 2
1	ExprTuple	66
2	Conditional	value: 3 condition: 4
3	Operation	operator: 13 operand: 7
4	Operation	operator: 31 operands: 6
5	ExprTuple	7
6	ExprTuple	66, 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: 58 body: 20
18	Operation	operator: 21 operand: 30
19	ExprTuple	58
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	58, 30
28	Operation	operator: 31 operands: 32
29	Operation	operator: 33 operands: 34
30	Variable
31	Literal
32	ExprTuple	40, 35
33	Literal
34	ExprTuple	36, 37
35	Variable
36	Operation	operator: 52 operands: 38
37	Operation	operator: 42 operand: 41
38	ExprTuple	58, 40
39	ExprTuple	41
40	Operation	operator: 42 operand: 45
41	Lambda	parameters: 62 body: 44
42	Literal
43	ExprTuple	45
44	Conditional	value: 46 condition: 50
45	Lambda	parameters: 62 body: 47
46	Operation	operator: 52 operands: 48
47	Conditional	value: 49 condition: 50
48	ExprTuple	51, 57
49	Operation	operator: 52 operands: 53
50	Operation	operator: 54 operands: 62
51	Operation	operator: 55 operands: 56
52	Literal
53	ExprTuple	59, 57
54	Variable
55	Literal
56	ExprTuple	58, 59
57	Operation	operator: 60 operands: 62
58	Variable
59	Operation	operator: 61 operands: 62
60	Variable
61	Variable
62	ExprTuple	63
63	ExprRange	lambda_map: 64 start_index: 65 end_index: 66
64	Lambda	parameter: 70 body: 67
65	Literal
66	Variable
67	IndexedVar	variable: 68 index: 70
68	Variable
69	ExprTuple	70
70	Variable