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, 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 = 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())

\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]

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