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, i, k
from proveit.core_expr_types import a_1_to_i
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 = [i], instance_expr = Forall(instance_param_or_params = [k, a_1_to_i], instance_expr = Equals(VecAdd(ExprRange(sub_expr1, ScalarMult(k, IndexedVar(a, sub_expr1)), one, i)), ScalarMult(k, VecAdd(a_1_to_i))).with_wrapping_at(2), domains = [K, V]), 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_{i \in \mathbb{N}}~\left[\forall_{k \in K,\left(a_{1} \in V\right), \left(a_{2} \in V\right), \ldots, \left(a_{i} \in V\right)}~\left(\begin{array}{c} \begin{array}{l} \left(\left(k \cdot a_{1}\right) +  \left(k \cdot a_{2}\right) +  \ldots +  \left(k \cdot a_{i}\right)\right) =  \\ \left(k \cdot \left(a_{1} +  a_{2} +  \ldots +  a_{i}\right)\right) \end{array} \end{array}\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: 42 body: 1
1	Operation	operator: 19 operand: 3
2	ExprTuple	3
3	Lambda	parameter: 51 body: 5
4	ExprTuple	51
5	Conditional	value: 6 condition: 7
6	Operation	operator: 19 operand: 11
7	Operation	operator: 9 operands: 10
8	ExprTuple	11
9	Literal
10	ExprTuple	51, 12
11	Lambda	parameter: 56 body: 14
12	Operation	operator: 15 operand: 42
13	ExprTuple	56
14	Conditional	value: 17 condition: 18
15	Literal
16	ExprTuple	42
17	Operation	operator: 19 operand: 22
18	Operation	operator: 47 operands: 21
19	Literal
20	ExprTuple	22
21	ExprTuple	56, 23
22	Lambda	parameters: 24 body: 25
23	Literal
24	ExprTuple	57, 50
25	Conditional	value: 26 condition: 27
26	Operation	operator: 28 operands: 29
27	Operation	operator: 30 operands: 31
28	Literal
29	ExprTuple	32, 33
30	Literal
31	ExprTuple	34, 35
32	Operation	operator: 45 operands: 36
33	Operation	operator: 52 operands: 37
34	Operation	operator: 47 operands: 38
35	ExprRange	lambda_map: 39 start_index: 55 end_index: 56
36	ExprTuple	40
37	ExprTuple	57, 41
38	ExprTuple	57, 42
39	Lambda	parameter: 61 body: 43
40	ExprRange	lambda_map: 44 start_index: 55 end_index: 56
41	Operation	operator: 45 operands: 46
42	Variable
43	Operation	operator: 47 operands: 48
44	Lambda	parameter: 61 body: 49
45	Literal
46	ExprTuple	50
47	Literal
48	ExprTuple	58, 51
49	Operation	operator: 52 operands: 53
50	ExprRange	lambda_map: 54 start_index: 55 end_index: 56
51	Variable
52	Literal
53	ExprTuple	57, 58
54	Lambda	parameter: 61 body: 58
55	Literal
56	Variable
57	Variable
58	IndexedVar	variable: 59 index: 61
59	Variable
60	ExprTuple	61
61	Variable