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, ExprRange, IndexedVar, K, Lambda, V, Variable, a, n, v
from proveit.core_expr_types import a_1_to_n
from proveit.linear_algebra import ScalarMult, VecAdd, VecSpaces
from proveit.logic import Equals, Forall, InClass
from proveit.numbers import Natural, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Lambda(V, Conditional(Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [a_1_to_n], instance_expr = Forall(instance_param_or_params = [v], instance_expr = Equals(ScalarMult(VecAdd(a_1_to_n), v), VecAdd(ExprRange(sub_expr1, ScalarMult(IndexedVar(a, sub_expr1), v), one, n))).with_wrapping_at(2), domain = V), domain = K), domain = Natural), 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_{n \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{n} \in K}~\left[\forall_{v \in V}~\left(\begin{array}{c} \begin{array}{l} \left(\left(a_{1} +  a_{2} +  \ldots +  a_{n}\right) \cdot v\right) =  \\ \left(\left(a_{1} \cdot v\right) +  \left(a_{2} \cdot v\right) +  \ldots +  \left(a_{n} \cdot v\right)\right) \end{array} \end{array}\right)\right]\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: 42 body: 2
1	ExprTuple	42
2	Conditional	value: 3 condition: 4
3	Operation	operator: 23 operand: 8
4	Operation	operator: 6 operands: 7
5	ExprTuple	8
6	Literal
7	ExprTuple	42, 9
8	Lambda	parameter: 55 body: 11
9	Operation	operator: 12 operand: 43
10	ExprTuple	55
11	Conditional	value: 14 condition: 15
12	Literal
13	ExprTuple	43
14	Operation	operator: 23 operand: 18
15	Operation	operator: 38 operands: 17
16	ExprTuple	18
17	ExprTuple	55, 19
18	Lambda	parameters: 49 body: 20
19	Literal
20	Conditional	value: 21 condition: 22
21	Operation	operator: 23 operand: 27
22	Operation	operator: 25 operands: 26
23	Literal
24	ExprTuple	27
25	Literal
26	ExprTuple	28
27	Lambda	parameter: 59 body: 30
28	ExprRange	lambda_map: 31 start_index: 54 end_index: 55
29	ExprTuple	59
30	Conditional	value: 32 condition: 33
31	Lambda	parameter: 62 body: 34
32	Operation	operator: 35 operands: 36
33	Operation	operator: 38 operands: 37
34	Operation	operator: 38 operands: 39
35	Literal
36	ExprTuple	40, 41
37	ExprTuple	59, 42
38	Literal
39	ExprTuple	58, 43
40	Operation	operator: 56 operands: 44
41	Operation	operator: 48 operands: 45
42	Variable
43	Variable
44	ExprTuple	46, 59
45	ExprTuple	47
46	Operation	operator: 48 operands: 49
47	ExprRange	lambda_map: 50 start_index: 54 end_index: 55
48	Literal
49	ExprTuple	51
50	Lambda	parameter: 62 body: 52
51	ExprRange	lambda_map: 53 start_index: 54 end_index: 55
52	Operation	operator: 56 operands: 57
53	Lambda	parameter: 62 body: 58
54	Literal
55	Variable
56	Literal
57	ExprTuple	58, 59
58	IndexedVar	variable: 60 index: 62
59	Variable
60	Variable
61	ExprTuple	62
62	Variable