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

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

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_{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)

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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	37, 30
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	14, 15
12	Operation	operator: 25 operands: 16
13	Operation	operator: 32 operands: 17
14	Operation	operator: 27 operands: 18
15	ExprRange	lambda_map: 19 start_index: 35 end_index: 36
16	ExprTuple	20
17	ExprTuple	37, 21
18	ExprTuple	37, 22
19	Lambda	parameter: 41 body: 23
20	ExprRange	lambda_map: 24 start_index: 35 end_index: 36
21	Operation	operator: 25 operands: 26
22	Variable
23	Operation	operator: 27 operands: 28
24	Lambda	parameter: 41 body: 29
25	Literal
26	ExprTuple	30
27	Literal
28	ExprTuple	38, 31
29	Operation	operator: 32 operands: 33
30	ExprRange	lambda_map: 34 start_index: 35 end_index: 36
31	Variable
32	Literal
33	ExprTuple	37, 38
34	Lambda	parameter: 41 body: 38
35	Literal
36	Variable
37	Variable
38	IndexedVar	variable: 39 index: 41
39	Variable
40	ExprTuple	41
41	Variable