Expression of type Forall

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, 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
from proveit.numbers import Natural, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [V], instance_expr = 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), 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())

\forall_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\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]\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 25 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 44 body: 4
3	ExprTuple	44
4	Conditional	value: 5 condition: 6
5	Operation	operator: 25 operand: 10
6	Operation	operator: 8 operands: 9
7	ExprTuple	10
8	Literal
9	ExprTuple	44, 11
10	Lambda	parameter: 57 body: 13
11	Operation	operator: 14 operand: 45
12	ExprTuple	57
13	Conditional	value: 16 condition: 17
14	Literal
15	ExprTuple	45
16	Operation	operator: 25 operand: 20
17	Operation	operator: 40 operands: 19
18	ExprTuple	20
19	ExprTuple	57, 21
20	Lambda	parameters: 51 body: 22
21	Literal
22	Conditional	value: 23 condition: 24
23	Operation	operator: 25 operand: 29
24	Operation	operator: 27 operands: 28
25	Literal
26	ExprTuple	29
27	Literal
28	ExprTuple	30
29	Lambda	parameter: 61 body: 32
30	ExprRange	lambda_map: 33 start_index: 56 end_index: 57
31	ExprTuple	61
32	Conditional	value: 34 condition: 35
33	Lambda	parameter: 64 body: 36
34	Operation	operator: 37 operands: 38
35	Operation	operator: 40 operands: 39
36	Operation	operator: 40 operands: 41
37	Literal
38	ExprTuple	42, 43
39	ExprTuple	61, 44
40	Literal
41	ExprTuple	60, 45
42	Operation	operator: 58 operands: 46
43	Operation	operator: 50 operands: 47
44	Variable
45	Variable
46	ExprTuple	48, 61
47	ExprTuple	49
48	Operation	operator: 50 operands: 51
49	ExprRange	lambda_map: 52 start_index: 56 end_index: 57
50	Literal
51	ExprTuple	53
52	Lambda	parameter: 64 body: 54
53	ExprRange	lambda_map: 55 start_index: 56 end_index: 57
54	Operation	operator: 58 operands: 59
55	Lambda	parameter: 64 body: 60
56	Literal
57	Variable
58	Literal
59	ExprTuple	60, 61
60	IndexedVar	variable: 62 index: 64
61	Variable
62	Variable
63	ExprTuple	64
64	Variable