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
from proveit.logic import Equals, Forall, InSet
from proveit.numbers import Natural, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Lambda(n, Conditional(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), InSet(n, Natural)))

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())

n \mapsto \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] \textrm{ if } n \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: 44 body: 2
1	ExprTuple	44
2	Conditional	value: 3 condition: 4
3	Operation	operator: 12 operand: 7
4	Operation	operator: 27 operands: 6
5	ExprTuple	7
6	ExprTuple	44, 8
7	Lambda	parameters: 38 body: 9
8	Literal
9	Conditional	value: 10 condition: 11
10	Operation	operator: 12 operand: 16
11	Operation	operator: 14 operands: 15
12	Literal
13	ExprTuple	16
14	Literal
15	ExprTuple	17
16	Lambda	parameter: 48 body: 19
17	ExprRange	lambda_map: 20 start_index: 43 end_index: 44
18	ExprTuple	48
19	Conditional	value: 21 condition: 22
20	Lambda	parameter: 51 body: 23
21	Operation	operator: 24 operands: 25
22	Operation	operator: 27 operands: 26
23	Operation	operator: 27 operands: 28
24	Literal
25	ExprTuple	29, 30
26	ExprTuple	48, 31
27	Literal
28	ExprTuple	47, 32
29	Operation	operator: 45 operands: 33
30	Operation	operator: 37 operands: 34
31	Variable
32	Variable
33	ExprTuple	35, 48
34	ExprTuple	36
35	Operation	operator: 37 operands: 38
36	ExprRange	lambda_map: 39 start_index: 43 end_index: 44
37	Literal
38	ExprTuple	40
39	Lambda	parameter: 51 body: 41
40	ExprRange	lambda_map: 42 start_index: 43 end_index: 44
41	Operation	operator: 45 operands: 46
42	Lambda	parameter: 51 body: 47
43	Literal
44	Variable
45	Literal
46	ExprTuple	47, 48
47	IndexedVar	variable: 49 index: 51
48	Variable
49	Variable
50	ExprTuple	51
51	Variable