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 ExprRange, IndexedVar, K, Lambda, V, Variable, a, n, x
from proveit.core_expr_types import x_1_to_n
from proveit.linear_algebra import ScalarMult, VecAdd, VecSpaces
from proveit.logic import Equals, Forall
from proveit.numbers import Natural, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Lambda(K, Forall(instance_param_or_params = [V], instance_expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [a], instance_expr = Forall(instance_param_or_params = [x_1_to_n], instance_expr = Equals(ScalarMult(a, VecAdd(x_1_to_n)), VecAdd(ExprRange(sub_expr1, ScalarMult(a, IndexedVar(x, sub_expr1)), one, n))).with_wrapping_at(2), domain = V), domain = K), domain = Natural), domain = 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())

K \mapsto \left[\forall_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\left[\forall_{n \in \mathbb{N}}~\left[\forall_{a \in K}~\left[\forall_{x_{1}, x_{2}, \ldots, x_{n} \in V}~\left(\begin{array}{c} \begin{array}{l} \left(a \cdot \left(x_{1} +  x_{2} +  \ldots +  x_{n}\right)\right) =  \\ \left(\left(a \cdot x_{1}\right) +  \left(a \cdot x_{2}\right) +  \ldots +  \left(a \cdot x_{n}\right)\right) \end{array} \end{array}\right)\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	parameter: 31 body: 1
1	Operation	operator: 27 operand: 3
2	ExprTuple	3
3	Lambda	parameter: 55 body: 5
4	ExprTuple	55
5	Conditional	value: 6 condition: 7
6	Operation	operator: 27 operand: 11
7	Operation	operator: 9 operands: 10
8	ExprTuple	11
9	Literal
10	ExprTuple	55, 12
11	Lambda	parameter: 58 body: 14
12	Operation	operator: 15 operand: 31
13	ExprTuple	58
14	Conditional	value: 17 condition: 18
15	Literal
16	ExprTuple	31
17	Operation	operator: 27 operand: 21
18	Operation	operator: 51 operands: 20
19	ExprTuple	21
20	ExprTuple	58, 22
21	Lambda	parameter: 61 body: 24
22	Literal
23	ExprTuple	61
24	Conditional	value: 25 condition: 26
25	Operation	operator: 27 operand: 30
26	Operation	operator: 51 operands: 29
27	Literal
28	ExprTuple	30
29	ExprTuple	61, 31
30	Lambda	parameters: 49 body: 32
31	Variable
32	Conditional	value: 33 condition: 34
33	Operation	operator: 35 operands: 36
34	Operation	operator: 37 operands: 38
35	Literal
36	ExprTuple	39, 40
37	Literal
38	ExprTuple	41
39	Operation	operator: 59 operands: 42
40	Operation	operator: 48 operands: 43
41	ExprRange	lambda_map: 44 start_index: 57 end_index: 58
42	ExprTuple	61, 45
43	ExprTuple	46
44	Lambda	parameter: 65 body: 47
45	Operation	operator: 48 operands: 49
46	ExprRange	lambda_map: 50 start_index: 57 end_index: 58
47	Operation	operator: 51 operands: 52
48	Literal
49	ExprTuple	53
50	Lambda	parameter: 65 body: 54
51	Literal
52	ExprTuple	62, 55
53	ExprRange	lambda_map: 56 start_index: 57 end_index: 58
54	Operation	operator: 59 operands: 60
55	Variable
56	Lambda	parameter: 65 body: 62
57	Literal
58	Variable
59	Literal
60	ExprTuple	61, 62
61	Variable
62	IndexedVar	variable: 63 index: 65
63	Variable
64	ExprTuple	65
65	Variable